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THE LIBRARY

THE UNIVERSITY
OF CALIFORNIA

LOS ANGELES

FLIGHT WITHOUT FORMULA

THE MECHANICS OF THE AERO-
PLANE. A Study of the Principles of
Flight. By COMMANDANT DUCHENE.
Translated from the French by JOHN H.
LEDEBOER, B.A., andT. O'B. HUBBARD.
With 98 Illustrations and Diagrams.
8vo. 8s. net.

FLYING : Some Practical Experiences.
By GUSTAV HAMEL and CHARLES C.
TURNER. With 72 Illustrations. 8vo.
I2s. 6d. net.

LONGMANS, GREEN, AND CO.,

LONDON, NEW YORK, BOMBAY.CALCUTTA, MADRAS

FLIGHT WITHOUT
FORMULA

SIMPLE DISCUSSIONS ON THE
MECHANICS OF THE AEROPLANE

COMMANDANT DUCHENE

OF THE FRENCH GENIE

TRANSLATED FROM THE FRENCH BY

JOHN H. LEDEBOER, B.A.

ASSOCIATE FELLOW, AERONAUTICAL SOCIETY; EDITOR
"AERONAUTICS"; JOINT- AUTHOR OF "THE AEROPLANE"
TRANSLATOR OF " THE MECHANICS OF THE AEROPLANE"

SECOND EDITION

LONGMANS, GREEN, AND CO.

39 PATERNOSTER ROW, LONDON

FOURTH AVENUE & 30™ STREET, NEW YORK

BOMBAY, CALCUTTA, AND MADRAS

1916

First Published . . July 1914
Type Reset . . October 1916

I (>

TRANSLATOR'S PREFACE

and equations are necessary evils ; they repre-
sent, as it were, the shorthand of the mathematician and the
engineer, forming as they do the simplest and most con-
venient method of expressing certain relations between
facts and phenomena which appear complicated when
dressed in everyday garb. Nevertheless, it is to be feared
that their very appearance is forbidding and strikes terror
to the hearts of many readers not possessed of a mathematical
turn of mind. However baseless this prejudice may be —
as indeed it is — the fact remains that it exists, and has hi
the past deterred many from the study of the principles of
the aeroplane, which is playing a part of ever -increasing
importance in the life of the community.

The present work forms an attempt to cater for this class
of reader. It has throughout been written in the simplest
possible language, and contains in its whole extent not a
single formula. It treats of every one of the principles of
flight and of every one of the problems involved in the
mechanics of the aeroplane, and this without demanding
from the reader more than the most elementary knowledge
of arithmetic. The chapters on stability should prove of
particular interest to the pilot and the student, containing
as they do several new theories of the highest importance
here fully set out for the first time.

In conclusion, I have to thank Lieutenant T. O'B.
Hubbard, my collaborator for many years, for his kind and
diligent perusal of the proofs and for many helpful
suggestions.

J. H. L.

968209

CONTENTS

CHAPTER I

PAGE

FLIGHT IN STILL AIR — SPEED 1

CHAPTER II

PLIGHT IN STILL AIR — POWER 16

CHAPTER III
PLIGHT IN STILL AIR — POWER (concluded) .... 35

CHAPTER IV

FLIGHT IN STILL AIR — THE POWER-PLANT . . . . 53

CHAPTER V

FLIGHT IN STILL AIR — THE POWER-PLANT (concluded) . . 70

CHAPTER VI

STABILITY IN STILL AIR — LONGITUDINAL STABILITY . . . 90

CHAPTER VII

STABILITY IN STILL AIR — LONGITUDINAL STABILITY (cotl-

cluded) ..... . . . . 115

CHAPTER VIII

STABILITY IN STILL AIR — LATERAL STABILITY . . .142

CHAPTER IX

STABILITY IN STILL AIR LATERAL STABILITY (concluded)

DIRECTIONAL STABILITY, TURNING . . . . . 161

CHAPTER X

THE EFFECT OF WIND ON AEROPLANES . . . . 183

Flight without Formulae

Simple Discussions on the Mechanics
of the Aeroplane

CHAPTER I
FLIGHT IN STILL AIR

SPEED

NOWADAYS everyone understands something of the main
principles of aeroplane flight. It may be demonstrated
in the simplest possible way by plunging the hand in
water and trying to move it at some speed horizontally,
after first slightly inclining the palm, so as to meet or
" attack " the fluid at a small " angle of incidence." It
will be noticed at once that, although the hand remains
very nearly horizontal, and though it is moved horizon-
tally, the water exerts upon it a certain amount of pressure
directed nearly vertically upwards and tending to lift the
hand.

This, in effect, is the principle underlying the flight of
an aeroplane, which consists in drawing through the air
wings or planes in a position nearly horizontal, and thus
employing, for sustaining the weight of the whole machine,
the vertically upward pressure exerted by the air on these
wings, a pressure which is caused by the very forward
movement of the wings.

Hence, the sustentation and the forward movement of
an aeroplane are absolutely interdependent, and the former

2 FLIGHT WITHOUT FORMULAS

can only be produced, in still air, by the latter, out of which
it arises.

But the entire problem of aeroplane flight is not solved
merely by obtaining from the " relative " air current which
meets the wings, owing to their forward speed, sufficient
lift to sustain the weight of the machine ; an aeroplane,
in addition, must always encounter the relative air current
in the same attitude, and must neither upset nor be thrown
out of its path by even a slight aerial disturbance. In
other words, it is essential for an aeroplane to remain in
equilibrium — more, in stable equilibrium.

This consideration clearly divides the study of aeroplane
flight in calm air into two broad, natural parts :

The study of lift and the study of stability.

These two aspects will be dealt with successively, and
will be followed by a consideration of flight in disturbed air.

First we will proceed to examine the lift of an aeroplane
in still air.

Following the example of a bird, and in accordance with
the results obtained by experiments with models, the wings
of an aeroplane are given a span five or six times greater
than then1 fore-and-aft dimension, or " chord," while they
are also curved, so that their lower surface is concave.* It
is desirable to give the wings a large span as compared
to the chord, in order to reduce as far as possible the
escape or leakage of the air along the sides ; while it
has the further advantage of playing an important part
in stability. Again, the camber of the wings increases their
lift and at the same time reduces their head-resistance or
" drag."

The angle of incidence of a wing or plane is the anerle.

* In English this curvature of the wing is generally known as the
" camber." On the whole, it would perhaps be more accurate to describe
the upper surface as being convex, since highly efficient wings have been
designed in which the camber is confined to the upper surface, the lower
surface being perfectly flat. — TRANSLATOR.

FLIGHT IN STILL AIR 3

expressed in degrees, made by the chord of the curve in
profile with the direction of the aeroplane's flight.

As stated above, the pressure of the air on a wing mov-
ing horizontally is nearly vertical, but only nearly. For,
though it lifts, a wing at the same time offers a certain
amount of resistance — known either as head-resistance or
drag * — which may well be described as the price paid for
the lift.

As the result of the research work of several scientists,
and of M. Eiffel in particular, with scale models, unit figures,
or " coefficients," have been determined which enable us to
calculate the amount of lift possessed by a given surface
and its drag, when moving through the air at certain angles
and at certain speeds.

Hereafter the coefficient which serves to calculate the
lifting-power of a plane will be simply termed the lift,
while that whereon the calculation of its drag is based will
be known as the drag.

M. Eiffel has plotted the results of his experiments in
diagrams or curves, which give, for each type of wing, the
values of the lift and drag corresponding to the various
angles of incidence.

The following curves are here reproduced from M. Eiffel's
work, and relate to :

A flat plane (fig. 1).

A slightly cambered plane, a type used by Maurice

Farman (fig. 2).
A plane of medium camber, adopted by Breguet

(fig. 3).

A deeply cambered plane, used by Bleriot on his
No. XI. monoplanes, cross-Channel type (fig. 4).

* The word " drag " is here adopted, in accordance with Mr Archibald
Low's suggestion, in preference to the more usual " drift," in order to
prevent confusion, and so as to preserve for the latter term its more
general, and certainly more appropriate meaning, illustrated in the ex-
pression " the drift of an aeroplane from its course in a side-wind," or
" drifting before a current." — TRANSLATOR.

4 FLIGHT WITHOUT FORMULA

These diagrams are so simple as to render further
explanation superfluous.

.0.00.

0.02 001 000

Drag. Drag.

FIG. 1. — Flat plane. FIG. 2.— Maurice Farman plane.

The calculation of the lifting-power and the head-resist-
ance produced by a given type of plane, moving through

FLIGHT IN STILL AIR

the air at a given angle of incidence and at a given speed,
is exceedingly simple. To obtain the desired result all that

J0.08

0.00

0.02 0:01

Drag.

FIG. 3.— Br6guet plane.

O.QO

0 02 0.01 0.00

Drag.
FIG. 4.— Bleriot XI. plane.

is needed is to multiply either the lift or the drag co-
efficients, corresponding to the particular angle of incidence,

6 FLIGHT WITHOUT FORMULA

by the area of the plane (hi square metres, or, if English
measurements are adopted, hi square feet) and by the square
of the speed, hi metres per second (or miles per hour).*

EXAMPLE. — A Bleriot monoplane, type No. XI., has an
area of 15 sq. m., and flies at 20 m, per second at an angle
of incidence of 7°. (1) What weight can its wings lift, and
(2) what is the power required to propel the machine ?

Referring to the curve in fig. 4, the lift of this particular
type of wing at an angle of 7° is 0-05, and its drag 0-0055.

Hence

T -f. . Square of

Lift. Area. the Speed.

0-05 x 15 x 400

gives the required value of the lifting-power, i.e. 300 kg.
Again

Drag. Are, *£™£

0-0055 x 15 x 400
gives the value of the resistance of the wings, i.e. 33 kg.

Let us for the present only consider the question of lift,
leaving that of drag on one side.

From the method of calculation shown above we may
immediately proceed to draw some highly important de-
ductions regarding the speed of an aeroplane. The fore-
and-aft equilibrium of an aeroplane, hi fact, as will be
shown subsequently, is so adjusted that the aeroplane can
only fly at one fixed angle of incidence, so long as the
elevator or stabiliser remains untouched. By means of the
elevator, however, the angle of incidence can be varied
within certain limits.

In the previous example, let the Bleriot monoplane be
taken to have been designed to fly at 7°. It has already
been shown that this machine, with its area of 15 sq. m.
and its speed of 72 km. per hour, will give a lifting-power
equal to 300 kg. Now, if this lifting-power be greater than

* Throughout this work the metric system will henceforward be
strictly adhered to. — TRANSLATOR.

FLIGHT IN STILL AIR 7

the weight of the machine, the latter will tend to rise ; if the
weight be less, it will tend to descend. Perfectly horizontal
flight at a speed of 72 km. per hour is only possible if the
aeroplane weighs just 300 kg.

In other words, an aeroplane of a given weight and a
given plane-area can only fly horizontally at a given angle
of incidence at one single speed, which must be that at
which the lifting-power it produces is precisely equal to
the weight of the aeroplane.

Now it has already been shown that the lifting-power
for a given angle of incidence is obtained by multiplying
the lift coefficient corresponding to this angle by the plane
area and by the square of the speed. This, therefore, must
also give us the weight of the aeroplane. It is clear that
this is only possible for one definite speed, i.e. when the
square of the speed is equal to the weight, divided by the
area multiplied by the inverse of the lift. And since the
weight of the aeroplane divided by its area gives the load-
ing on the planes per sq. m., the following most important
and practical rule may be laid down :

The speed (in metres per second) of an aeroplane, flying
at a given angle of incidence, is obtained by multiplying
the square root of its loading (in kg. per sq. in.} by the square
root of the inverse of the lift corresponding to the given
angle.

At first sight the rule may appear complicated. Actually
it is exceedingly simple when applied.

EXAMPLE. — A Breguet aeroplane, with an area of 30 sq. m.
and weighing 600 kg., flies with a lift of 0-04, equivalent
(according to the curve in fig. 3) to an angle of incidence
of about 4°. What is its speed ?

600
The loading is ——=20 kg. per sq. m.

Square root of the loading— 4 '47.

Inverse of the lift is —=25.
0-04

Square root of inverse of the lift =5.

FLIGHT WITHOUT FORMULA

The speed required, therefore, in metres per second— 4-47
X 5=22- 3 m. per second, or about 80 km. per hour. But
if a different angle of incidence, or a different figure for
the lift — which is equivalent, and, as will be seen here-
after, more usual — be taken, a different speed will be
obtained.

Hence each angle of incidence has its own definite speed.

For instance, if we take the Breguet aeroplane already
considered, and calculate its speed for a whole series of
angles of incidence, we obtain the results shown in Table I.
But before examining these results in greater detail, so far
as the relation between the angles of incidence, or the lift,
and the speed is concerned, a few preliminary observations
may be useful.

TABLE I.

'Speed.

In m.p.s.

In km.p.h.

Lift.

Correspond-
ing Angle of
Incidence.

! Square
Inverse of i Root of
Lift. ; Inverse of
Lift.

~* >j~o 50

10 >»
|1^ .

J'ilfcr

lit"

:(? S 3 "8

|*a

0-020

0° (about)

7-07

31-6

113-6

0-030

2° ,,

33-3 577

25-8

92-8

0-040

4° ,,

5-00

22-3

80-3

0-050

64° „

4-47

20-0

72-0

0-060

10° ,,

16-6 4-08

18-2

65-6

0-066

15° ,,

15-2

3-90

17-4

62-6

In the first place, it should be noted that when the
Breguet whig has no angle of incidence, when, that is, the
wind meets it parallel to the chord, it still has a certain
lift. This constitutes one of the interesting properties of

FLIGHT IN STILL AIR 9

a cambered plane. While a flat plane meeting the air
edge-on has no lift whatever, as is evident, a cambered
plane striking the air in a direction parallel to its chord
still retains a certain lifting-power which varies according
to the plane section.

Thus, in those conditions a Breguet wing still has a lift of
0-019, and if figs. 4 and 2 are examined it will be seen that
at zero incidence the Bleriot No. XI. would similarly have
a lift of 0-012, but the Maurice Farman of only 0-006. It
follows that a cambered plane exerts no lift whatever only
when the wind strikes it slightly on the upper surface. In
other words, by virtue of this property, a cambered plane
may be regarded as possessing an imaginary chord — if the
expression be allowed — inclined at a negative angle (that is,
in the direction opposed to the ordinary angle of incidence)
to the chord of the profile of the plane viewed in section.

If the necessary experiments were made and the curves
on the diagrams were continued to the horizontal axis, it
would be found that the angle between this " imaginary
chord " and the actual chord is, for the Maurice Farman
plane section about 1°, for that of the Bleriot XI. some 2°,
and for that of the Breguet 4°.

Let it be noted in passing that in the case of nearly
every plane section a variation of 1° in the angle of incidence
is roughly equivalent to a variation in lift of 0-005, at any
rate for the smaller angles. One may therefore generalise
and say that for any ordinary plane section a lift of 0-015
corresponds to an angle of incidence of 3° relatively to the
" imaginary chord," a lift of 0-020 to an angle of 4°, a lift
of 0-025 to 5°, and so forth.

Turning now to the upper portion of the curves in the
diagrams, it will be seen that, beginning with a definite
angle of incidence, usually in the neighbourhood of 15°, the
lift of a plane no longer increases. The curves relating to
the Breguet and the Bleriot cease at 15°, but the Maurice
Farman curve clearly shows that for angles of incidence
greater than 15° the lift gradually diminishes. Such coarse

10 FLIGHT WITHOUT FORMULA

angles, however, are never used in practice, for a reason
shown in the diagrams, which is the excessive increase in
the drag when the angle of incidence is greater than 10°.
In aviation the angles of incidence that are employed there-
fore only vary within narrow limits, the variation certainly
not surpassing 10°.

We may now return to the main object for which Table I.
was compiled, namely, the variation in the speed of an
aeroplane according to the angle of incidence of its planes.

First, it is seen that speed and angle of incidence vary
inversely, which is obvious enough when it is remembered
that in order to support its own weight, which necessarily
remains constant, an aeroplane must fly the faster the
smaller the angle at which its planes meet the air.

Secondly, it will be seen that the variation in speed is
more pronounced for the smaller angles of incidence ; hence,
by utilising a small lift coefficient great speeds can be
attained. Thus, for a lift equal to 0-02, at which the
Breguet wing would meet the air along its geometrical
chord, the speed of the aeroplane, according to Table I.,
would exceed 113 km. an hour.

If an aeroplane could fly with a lift coefficient of O'Ol,
that is, if the planes met the air with their upper surface —
the imaginary chord would then have an angle of incidence
of no more than 2° — the same method of calculation would
give a speed of over 160 km. per hour.

The chief reason which in practice places a limit on the
reduction of the lift is, as will be shown subsequently, the
rapid increase in the motive -power required to obtain high
speeds with small angles of incidence. And further, there
is a considerable element of danger in unduly small angles.
For instance, if an aeroplane were to fly with a lift of O'Ol
— so that the imaginary chord met the air at an angle of
only 2° — a slight longitudinal oscillation, only just exceeding
this very small angle, would be enough to convert the fierce
air current striking the aeroplane moving at an enormous
speed from a lifting force into one provoking a fall. It is

FLIGHT IN STILL AIR 11

true that the machine would for an instant preserve its
speed owing to inertia, but the least that could happen
would be a violent dive, which could only end in disaster
if the machine was flying near the ground.

Nevertheless there are certain pilots, to whom the word
intrepid may be justly applied, who deny the danger and
argue that the disturbing oscillation is the less likely to
occur the smaller the angle of incidence, for it is true, as
will be seen hereafter, that a small angle of incidence is an
important condition of stability. However this may be,
there can be no question but that flying at a very small
angle of incidence may set up excessive strains in the frame-
work, which, in consequence, would have to be given
enormous strength . Thus , if it were possible for an aeroplane
to fly with a lift coefficient of 0-01, and if, owing to a wind
gust or to a manoeuvre corresponding to the sudden " flatten-
ing out " practised by birds of prey and by aviators at the
conclusion of a dive, the plane suddenly met the air at an
angle of incidence at which the lift reaches a maximum —
that is, from 0-06 to 0-07 according to the type of plane —
the machine would have to support, the speed remaining
constant for the time being by reason of inertia, a pressure
six or seven times greater than that encountered in normal
flight, or than its own weight.

In practice, therefore, various considerations place a limit
on the decrease of the angle of incidence, and it would
accordingly appear doubtful whether hitherto an aeroplane
has flown with a lift coefficient smaller than 0-02.*

It is easy enough to find out the value of the lift co-
efficient at which exceptionally high speeds have been
attained from a few known particulars relating to the
machine in question. The particulars required are :

The velocity of the aeroplane, which must have been
carefully timed and corrected for the speed of the wind ;

The total weight of the aeroplane fully loaded ;

The supporting area.

* See footnote on p. 12.

12 FLIGHT WITHOUT FORMULA

The lift may then be found by dividing the loading of
the planes by the square of the speed in metres per second.

EXAMPLE. — An aeroplane with a plane area of 12 sq. m.
and weighing, fully loaded, 360 kg. has flown at a speed of
130 km. or 36-1 m. per second. What was its lift coefficient ?

O f*(\

The loading= = 30 kg. per sq. m.

Square of the speed=1300.

Ofi

Required lift=-?"- =about 0-023.*

Table I. further shows that when the angle of incidence
reaches the neighbourhood of 15° (which cannot, as has
been seen, be employed in practical flight) the lift reaches
its maximum value, and the speed consequently its minimum.

* At the time of writing (August 1913) the speed record, 171 '7 km.
per hour or 47'6 m. per second, is held by the Deperdussin monocoque
with a 140-h.p. motor, weighing 525 kg. with full load, and with a plane
area of about 12 sq. m. (loading, 43'7 kg. per sq. m.). Another machine
of the same type, but with a 100-h.p. engine, weighing 470 kg. in all, and
with an area of 11 sq. m., has attained a speed of 168 km. per hour or
46'8 m. per second. According to the above method of calculation, the
flight in both cases was made with a lift coefficient of about 0'0195.
— AUTHOR.

Since the above was written, all speed records were broken during
the last Gordon-Bennett race in September 1913. The winner was
Prevost, on a 160-h.p. Gnome Deperdussin monoplane, who attained a
speed of a fraction under 204 km. per hour ; while Vedrines, on a 160-
h.p. Gnome-Ponnier monoplane, achieved close upon 201 km. per hour.
The Deperdussin monoplane, with an area of 10 sq. m., weighed, fully
loaded, about 680 kg. ; the Ponnier, measuring 8 sq. m., weighed ap-
proximately 500 kg. Adopting the same method of calculation, it is
easily shown that the lift coefficients worked out at 0'021 and 0'020
respectively. It is just possible that these figures were actually slightly
smaller, since it is difficult to determine the weights with any consider-
able degree of accuracy. However, the error, if there be any, is only
slight, and the result only confirms the author's conclusions. Since that
time Emile Vedrines is stated to have attained, during an official trial,
a speed of 212 km. per hour, on a still smaller Ponnier monoplane
measuring only 7 sq. m. in area and weighing only 450 kg. in flight.
This would imply a lift coefficient of 0'0185, a figure which cannot be
accepted without reserve. — TRANSLATOR.

FLIGHT IN STILL AIR 13

If the angle surpassed 15° the lift would diminish and the
speed again increase.

A given aeroplane, therefore, cannot in fact fly below a
certain limit speed, which in the case of the Breguet already
considered, for instance, is about 63 km. per hour.

It will be further noticed that in Table I. one of the
columns, the second one, contains particulars relating only
to the Breguet type of plane. If this column were omitted,
the whole table would give the speed variation of any
aeroplane with a loading of 20 kg. per sq. m. on its
planes, for a variation in the lift coefficient of the planes.
It was this that led to the above remark, made in passing,
that it was more usual to take the lift coefficient than the
angle of incidence ; for the former is independent of the
shape of the plane.

The speed variation of an aeroplane for a variation in its
lift coefficient can easily be plotted hi a curve, which would
have the shape shown in fig. 5, which is based on the figures
in Table I.

The previous considerations relate more especially to a
study of the speeds at which a given type of aeroplane can
fly. In order to compare the speeds at which different
types of aeroplanes can fly at the same lift coefficient, we
need only return to the basic rule already set forth (p. 7). .
It then becomes evident that these speeds are to one
another as the square roots of the loading.

The fact that only the loading comes into consideration
in calculating the speed of an aeroplane shows that the
speed, for a given lift coefficient, of a machine does not
depend on the absolute values of its weight and its plane
area, but only on the ratio of these latter. The most heavily
loaded aeroplanes yet built (those of the French military
trials in 1911) were loaded to the extent of 40 kg. per
sq. m. of plane area.* The square root of this number
being 6-32, an aeroplane of this type, driven by a sufficiently

* The 140-kp. Deperdussin monocoque had a loading of 43-8 kg.

per sq. m.

FLIGHT WITHOUT FORMULA

powerful engine to enable it to fly at a lift coefficient of
0-02 (the square root of whose inverse is 7-07), could have
attained a speed equal to 6-32 x 7-07, that is, it could have
exceeded 44-5 m. per second or 160 km. per hour.

It is therefore evident that there are only two means for
increasing the speed of an aeroplane — either to reduce the

Lift

0.0/0 0.020 0.030 0,0*0 0.050 0.060 0.076
FIG. 5.

lift coefficient or to increase the loading. Both methods
require power ; we shall see further on which of the two is
the more economical in this respect.

The former has the disadvantages — contested, it is true
—which have already been stated. The latter requires
exceptionally strong planes.

In any case, it would appear that, in the present stage of
aeroplane construction, the speed of machines will scarcely
exceed 150 to 160 km. per hour ; and even so, this result

FLIGHT IN STILL AIR 15

could only have been achieved with the aid of good engines
developing from 120 to 130 h.p.* So that we are still far
removed from the speeds of 200 and even 300 km. per hour
which were prophesied on the morrow of the first advent
of the aeroplane, f

In concluding these observations on the speed of aero-
planes, attention may be drawn to a rule already laid down
in a previous work,! which gives a rapid method of calculat-
ing with fair accuracy the speed of an average machine
whose weight and plane area are known.

The speed of an average aeroplane, in metres per second,
is equal to five times the square root of its loading, in kg.
per sq. m.

This rule simply presupposes that the average aeroplane
flies with a lift coefficient of 0-04, the inverse of whose
square root is 5. The rule, of course, is not absolutely
accurate, but has the merit of being easy to remember and
to apply.

EXAMPLE. — What is the speed of an aeroplane weighing
900 kg., and having an area of 36 sq. m. ?

900
Loading = — r = 25 kg. per sq. m.

Square root of the loading =5.

Speed required=5x5=25 m. per second or 90 km.
per hour.

* In previous footnotes it has already been stated that the Deper-
dussin monocoques, a 140-h.p. and a 100-h.p., have already flown at
about 170 km. per hour. But these were exceptions, and, on the whole,
the author's contention remains perfectly accurate even to-day. — TRANS-
LATOR.

f The reference, of course, is only to aeroplanes designed for everyday
use, and not to racing machines. — TRANSLATOR.

J The Mechanics of the Aeroplane (Longmans, Green & Co.).

CHAPTER II

FLIGHT IN STILL AIR

POWER

IN the first chapter the speed of the aeroplane was dealt
with in its relation to the constructional features of the
machine, or its characteristics (i.e. the weight and plane
area), and to its angle of incidence. It may seem strange
that, in considering the speed of a motor-driven vehicle, no
account should have been taken of the one element which
usually determines the speed of such vehicles, that is, of
the motive-power. But the anomaly is only apparent, and
wholly due to the unique nature of the aeroplane, which
alone possesses the faculty — denied to terrestrial vehicles
which are compelled to crawl along the surface of the
earth, or, in other words, to move hi but two dimensions — of
being free to move upwards and downwards, in all three
dimensions, that is, of space.

The subject of this chapter and the next will be to
examine the part played by the motive-power in aeroplane
flight, and its effect on the value of the speed.

In all that has gone before it has been assumed that, in
order to achieve horizontal flight, an aeroplane must be
drawn forward at a speed sufficient to cause the weight of
the whole machine to be balanced by the lifting power
exerted by the planes. But hitherto we have left out of
consideration the means whereby the aeroplane is endowed
with the speed essential for the production of the necessary
lifting-power, and we purposely omitted, at the time, to

FLIGHT IN STILL AIR 17

deal with the head-resistance or drag, which constitutes, as
already stated, the price to be paid for the lift.

This point will now be considered.

Reverting to the concrete case first examined, that of the
horizontal flight at an angle of incidence of 7° of a Bleriot
monoplane weighing 300 kg. and possessing a wing area of
15 sq. m., it has been seen that the speed of this machine
flying at this angle would be 20 m. per second or 72 km.
per hour, and that the drag of the wings at the speed
mentioned would amount to 33 kg.

Unfortunately, though alone producing lift in an aero-
plane, the planes are not the only portions productive of
drag, for they have to draw along the fuselage, or inter-
plane connections, the landing chassis, the motor, the
occupants, etc.

For reasons of simplicity, it may be assumed that all these
together exert the same amount of resistance or drag as
that offered by an imaginary plate placed at right angles
to the wind, so as to be struck full in the face, whose area
is termed the detrimental surface of the aeroplane.

M. Eiffel has calculated from experiments with scale
models that the detrimental surface of the average
single-seater monoplane amounted to between f and 1
sq. m., and that of an average large biplane to about
1| sq. m.* But it is clear that these calculations can
only have an approximate value, and that the detrimental
surface of an aeroplane must always be an uncertain
quantity.

But in any case it is evident that this parasitical effect
should be reduced to the lowest possible limits by stream-
lining every part offering head-resistance, by diminishing
exterior stay wires to the utmost extent compatible with
safety, etc. And it will be shown hereafter that these
measures become the more important the greater the
speed of flight.

The drag or passive resistance can be easily calculated
* These figures have since been undoubtedly reduced.

18 PLIGHT WITHOUT FORMULA

for a given detrimental surface by multiplying its area
in square metres by the coefficient 0-08 (found to be the
average from experiments with plates placed normally), and
by the square of the speed in metres per second.

Thus, taking once again the Bleriot monoplane, let us
suppose it to possess a detrimental surface of 0-8 sq. m. ;
its drag at a speed of 72 km. per hour or 20 m. per second
will be:

n «? • * Detrimental Square of
Coefficient. gurface> ^ gpeed>

0-08 x 0-8 x 400 =26 kg. (about).

As the drag of the planes alone at the above speed
amounts to 33 kg., it is necessary to add this figure of
26 kg., in order to find the total resistance, which is
therefore equal to 59 kg. The principles of mechanics
teach that to overcome a resistance of 59 kg. at a speed
of 20 m. per second, power must be exerted whose amount,
expressed in horse-power, is found by dividing the product
of the resistance (59 kg.) and the speed (20 m. per second)
by 75.* We thus obtain 16 h.p. But a motor of 16 h.p.
would be insufficient to meet the requirements.

I^or the propelling plant, consisting of motor and pro-
peller, designed to overcome the drag or air resistance of
the aeroplane, is like every other piece of machinery subject
to losses of energy. Its efficiency, therefore, is only a
portion of the power actually developed by the motor.
The efficiency of the power-plant is the ratio of useful,
power — that is, the power capable of being turned to effect
after transmission — to the motive power.

Thus, in order to produce the 16 h.p. required for
horizontal flight in the above case of the Bleriot mono-

* This is easily understood. The unit of power, or horse-power, is
the power required to raise a weight of 75 kg. to a height of 1 in. in
1 second, so that, to raise in this time a weight of 59 kg. to a height of

59 x 20
20 m., we require — =^ — h.p. Exactly the same holds good if, instead

of overcoming the vertical force of gravity, we have to overcome the
horizontal resistance of the air.

FLIGHT IN STILL AIR 19

plane, it would be necessary to possess an engine develop-
ing 32 h.p. if the efficiency is only 50 per cent., 26-6 h.p.
for an efficiency of 60 per cent., etc.

But if the aeroplane were to fly at an angle of incidence
other than 7° — which, as already stated, would depend on
the position of the elevator — the speed would necessarily
be altered. If this primary condition were modified, the
immediate result would be a variation in the drag of the
planes, in the head-resistance of the aeroplane, in the
propeller-thrust, which is equal to the total drag, and lastly,
in the useful power required for flight.

Each value of the angle of incidence — and consequently
of the speed — therefore has only one corresponding value
of the useful power necessary for horizontal flight.

Returning to the Breguet aeroplane weighing 600 kg.
with a plane area of 30 sq. m., on which Table I. was based,
we may calculate the values of the useful powers required
to enable it to fly along a horizontal path for different
angles of incidence and for different lifts. The detrimental
surface may be assumed, for the sake of simplicity, to be
1-2 sq. m.

The values of the drag corresponding to those of the
lift will be obtained from the polar diagram shown in
fig. 3.

Table II., p. 20, summarises the results of the calcula-
tion required to find the values of the useful powers for
horizontal flight at different lift coefficients.

Various and interesting conclusions may be drawn from
the figures in columns 8 and 9 of this Table.

In the first place, it will be noticed from the figures in
column 8 that the propeller-thrust (equivalent to the drag
of the planes added to the head-resistance of the machine,
i.e. column 6 and column 7) has a minimum value of 91 kg.,
corresponding to a lift coefficient of 0-05, and to the angle
6£°. This angle, which, in the case under consideration,
is that corresponding to the smallest propeller-thrust, is
usually known as the optimum angle of the aeroplane.

FLIGHT WITHOUT FORMULA

TABLE II.

•« «

Speed Values.

Bfe

g^g

§1^' -S
g g ^|| g

it-is

*-:-£

&C ^"

3 ^ o 5T1

o ~~ '**. ^a

Lift.

ic
s *s

o |

Drag of P
(drag x area, 3
x square of

il«^

llfi

|sl i1

wsl x

Propeller- T

Useful Power f
zontal Flight (
product of figs
3 and 8 divide

m.p.s.

kni.p.h.

0-020

0°

31-6

113-6

0-0022

66kg.

96kg.

162kg.

0-030

2°

25-8

92-8

0-0024! 48

112

0-040

4°

22-3

80-3

0-0032

0-050

64*

20-0

72-0

0-0044

0-060

10°

18-2

65-6

0-0063

0-066

15°

17-4 62-6

0-0118

107

136

When the lift coefficient is small, the requisite thrust,
it will be seen, increases very rapidly, and the same holds
good for high lift coefficients.

Secondly, the figures in column 9 show that, together
with the thrust, the useful power required for flight reaches
a minimum of 23 h.p., corresponding to a lift value of
0-06. The angle of incidence at which this minimum of
useful power can be achieved, about 10° in the present case,
can be termed the economical angle.

This angle is greater than the optimum angle, which can
be explained by the fact that, though the thrust begins
to increase again, albeit very slowly, when the angle of
incidence is raised above the optimum angle, the speed still
continues to decrease to an appreciable extent, and for the
time being this decrease in speed affects the useful power
more strongly than the increasing thrust ; and the minimum
value of the useful power is, consequently, not attained
until, as the angle of incidence continues to grow, the

FLIGHT IN STILL AIR 21

increase in the thrust exactly balances the decrease in the

The figures in column 9 again show the great expenditure
of power required for flight at a low lift coefficient. Thus,
the Breguet aeroplane already referred to, driven by a
propelling plant of 50 per cent, efficiency, flying at a lift of
0-05 — that is, at a speed of 72 km. per hour — only requires
an engine developing 46 h.p. ; but it would need a 136-h.p.
engine to fly with a lift of O02, or at about 113 km. per
hour. It is mainly on this account that, as we have already
stated, the use of low lift coefficients is strictly limited.

The variations in power corresponding to variations in
speed (and in lift) can be plotted in a simple curve.

Fig. 6 is of exceptional importance, for it may be said to
determine the character of the machine, and will hereafter
be referred to as the essential aeroplane curve.

After these preliminary considerations on the power
required for horizontal flight, we may now proceed to
examine the precise nature of the effect of the motive-
power on the speed, which will lead at the same time to
certain conclusions relating to gliding flight *

For this, recourse must be had to one of the most elemen-
tary principles of mechanics, known as the composition
and decomposition of forces. The principle is one which
is almost self-evident, and has, in fact, already been used in
these pages, when at the beginning of Chapter I. it was
shown that in the air pressure, which is almost vertical, on
a plane moving horizontally, a clear distinction must be
made between the principal part of this pressure, which is
strictly vertical (the lift), and a secondary part, which is
strictly horizontal (the drag).

And, conversely, it is evident that for the action of two
forces working together at the same time may be substituted

* There is really no excuse for the importation into English of the
French term " vol plane," and still less for the horrid anglicism
" volplane," since " gliding flight " is a perfect English equivalent of the
French. — TRANSLATOR.

FLIGHT WITHOUT FORMULA

that of a single force, termed the resultant of these two
forces. This proceeding is known as the composition of
forces. So, in compounding the vertical reaction con-
stituting the lift, and the horizontal reaction which forms

71)

Flying Speeds (m/s).

FIG. 6.

The figures on the curve indicate the lift.

the drift, one obtains the total air pressure, which is simply
their resultant.

Both the composition and decomposition of forces is
accomplished by way of projection. Thus (fig. 7), the force
Q,* which is inclined, can be decomposed into two forces,

* A force is represented by a straight line, drawn in the direction in
which the force operates, and of a length just proportional to the
magnitude of the force.

FLIGHT IN STILL AIR

F and r, vertical and horizontal respectively, by projecting
in the horizontal and vertical directions the end point A on
two axes starting from the point 0, where the forces are
applied. Conversely, these two forces F and r may be

FIG. 7.

FligU-Path. ._

FIG. 8.

compounded into one resultant Q, by drawing the diagonal
of the parallelogram or rectangle of which they form two
of the sides. We may now return to the problem under
consideration.

If we take the aeroplane as a whole, instead of dealing
with the planes alone, it will be readily seen that the
horizontal component of the air pressure on the whole

24 FLIGHT WITHOUT FORMULA

machine is equal to the drag of the planes added to the
passive or head-resistance, the while the vertical component
remains practically equal to the bare lift of the planes,
since the remaining parts of the structure of an aeroplane
exert but slight lift, if at all.* The entire pressure of the
air on a complete aeroplane in flight is therefore farther
inclined to the perpendicular than that exerted on the planes
alone.

If (see fig. 8) the aeroplane is assumed to be represented
by a single point O, in horizontal flight, the air pressure Q
exerted upon it may be decomposed into two forces, of
which the lift F is equal and directly opposite to the
weight P, and the drag r, or total resistance to forward
movement, which must be exactly balanced by the thrust
t of the propeller.

But, supposing the engine be stopped and the propeller
consequently to produce no thrust (fig. 9), the aeroplane
will assume a descending flight-path such that the planes
still retain the single angle of 7°, for instance, which we
have assumed, so long as the elevator is not moved, and
such that the air pressure Q on the planes becomes
absolutely vertical, in order to balance the weight of the
machine, instead of remaining inclined as heretofore. This
is gliding flight.

Relatively to the direction of flight, the air pressure Q
still retains its two components, of which r is simply the
resistance of the air opposed to the forward movement of
the glider. The second component F is identical to the
lifting power in horizontal flight, and its value is obtained
by multiplying the lift coefficient corresponding to the
angle 7° by the plane area, and by the square of the speed
of the aeroplane on its downward flight-path.

Fig. 9 shows that, by the very fact of being inclined, the
force F is slightly less than the weight of the machine, but,
since the gliding angle of an aeroplane is usually a slight

* For the sake of simplicity, we may consider that the tail plane,
which will be hereinafter dealt with, exerts no lift.

FLIGHT IN STILL AIR

one, the lifting power F may still be deemed to be equal to
the weight of the aeroplane.

Clearly, therefore, every consideration in the first chapter
which related to the speed in horizontal flight is equally
applicable to gliding flight, so that it may be said that

FIG. 9.

when an aeroplane begins to glide, without changing its
angle, the speed remains the same as before.

In fact, horizontal flight is simply a glide in which the
angle of the flight-path has been raised by mechanical
means.

On comparing figs. 8 and 9 it will be seen that this angle
is that which, in fig. 8, is marked by the letter p. If this

FLIGHT WITHOUT FORMULAE

angle is represented, as in the case of any gradient, in
terms of a decimal fraction, it will be found to depend on
the ratio which the forces r and F bear to one another.
Hence, the following rule may be stated : —

RULE. — The gliding angle assumed at a given angle of
incidence by any aeroplane is equal to the thrust required
for its horizontal flight at the same angle, divided by the
weight of the machine.

Thus the Bleriot monoplane dealt with in the first
instance, which requires for horizontal flight at an angle
of 7° a thrust of 59 kg., and weighs 300 kg., would assume
on its glide, at the same angle of incidence, a descending

flight-path equal to — — , or 0-197, which is equivalent to
oOO

nearly 20 cm. in every metre (1 in 5). The Breguet
aeroplane on which Tables I. and II. were based, weighing
600 kg., would assume at different angles (or lift coefficients)
the gliding angles shown in Table III.

TABLE III.

Angle
corre-
Lift. spending
to the
Lift.

Speed.

Propeller-
Thrust in
Horizontal

Flight.

Gliding Angle
Weight
(600kg.)
divided by
figures in
col. 5.

m/s.

km/li.

1 2

3 4

0-02 0°

31-6

113-6

162

0-270

0-03 | 2°

25-8

92-8

112

0-187

0-04 ! 4°

22-3

80-3

0-160

6i°

20-0

72-0 91

0-151

0-06 10°

18-2

65-6

9f>

0-158

0-066 15°

17-4

62-6

136

0-22C

It will now be seen that the best gliding angle is obtained
when the angle of incidence is the same as the optimum

FLIGHT IN STILL AIR 27

angle of the aeroplane. The latter, therefore, is the best
from the gliding point of view, so far as the length of the
glide is concerned.

In fig. 10, starting from a point 0, are drawn dotted lines
corresponding to the gliding angle given in column 6 of
Table III., and on these lines are marked off distances
proportional to the speed values set out in columns 3 or 4 ;
the diagram will then give, if the points are connected
into a curve, the positions assumed, in unit time, by a
glider, launched at the various angles from the point 0.

It will be observed in the first place that any given
gliding path, such as OA, for instance, cuts the curve at
two points, A and B, thus showing that this gliding path
could have been traversed by the aeroplane at two different
speeds, OA and OB, corresponding to the two different
angles of incidence, 1° and 15° in the present case.

Only for the single gliding path OM, corresponding to
the smallest gliding slope and the optimum angle of inci-
dence, do these two points coincide.

But it is not by following this gliding path that an aero-
plane will descend best in the vertical sense during a given
period of time ; for this it will only do by following the
path OE corresponding to the highest point on the curve,
and the angle of incidence to be adopted to achieve this
result is none other than the economical angle. But the
difference in the rate of fall is only slight for the example
in question.

It will be noted that as the angle of incidence diminishes,
the gliding angle rapidly becomes steeper. If the curve
were extended so as to take in very small angles of
incidence, it would be found that at a lift coefficient of
0-015 the gliding path would already have become very
steep, that this steepness would increase very rapidly for
the coefficient 0-010, and that at 0-005 it approached a
headlong fall. The fall, in fact, must become vertical when
the lift disappears, that is, when the plane meets the air
along its imaginary chord.

28 FLIGHT WITHOUT FORMULA

FLIGHT IN STILL AIR 29

In these conditions, a slight variation in the lift there-
fore brings about a very large alteration in the gliding
angle, and this effect is the more intense the smaller the
lift coefficient. The glide becomes a dive. Hence it is
clear that this is another danger of adopting a low lift
coefficient.

This brief discussion on gliding flight, interesting enough
in itself, was necessary to a proper understanding of the
part played by power in the horizontal flight of an aero-
plane, for we can now regard the latter in the light of
a glide in which the gliding path has been artificially
raised.

And this raising of the gliding path is due to the power
derived from the propelling plant.

This will be better understood if we assume that, during
the course of a glide, the pilot started up his engine again
without altering the position of the elevator, so that the
planes remained at the same angle as before ; the gliding
path would gradually be raised until it attained and even
surpassed the horizontal, while the aeroplane (as has been
seen) would approximately maintain the same speed
throughout.

Hence it may be said that when the angle of incidence
remains constant, the speed of an aeroplane is not produced
by its motive power, as in the case of all other existing
vehicles, since, when the motor is stopped, this speed is
maintained.

The function of the power-plant is simply to overcome
gravity, to prevent the aeroplane from yielding, as it in-
evitably must do in calm air, to the attraction of the earth ;
in other words, to govern its vertical flight.

In the case now under consideration, the speed therefore
is wholly independent of the power, since, as has been seen,
it is entirely determined by the angle of incidence, and if
this remains constant, as assumed, any excess of power will
simply cause the aeroplane to climb, while a lack of power

30 FLIGHT WITHOUT FORMULA

will cause it to coine down, but without any variation in the
speed.

But this must not be taken to imply that the available
motive-power cannot be transformed into speed, for such,
happily, is not the case. Hitherto the elevator has been
assumed to be immovable so that the incidence remained
constant.

As a matter of fact, the incidence need only be diminished
through the action of the elevator in order to enable the
aeroplane to adopt the speed corresponding to the new
angle of planes, and in this way to absorb the excess of
power without climbing.

Nevertheless — and the point should be insisted upon as
it is one of the essential principles of aeroplane flight — the
angle of incidence alone determines the speed, which cannot
be affected by the power save through the intermediary of
the incidence.

Hitherto we have constantly alluded to the different
speeds at which an aeroplane can fly, as if, in practice,
pilots were able to drive their machines at almost any
speed they desired. In actual fact, a given aeroplane
usually only flies at a single speed, so that we are in
the habit of referring to the X biplane as a 70 km.
per hour machine, or of stating that the Y monoplane
does 100 km. per hour. This is simply because up to
now, and with very few exceptions, pilots run their engines
at their normal number of revolutions. In these conditions
it is evident that the useful power furnished by the
propelling plant determines the incidence, and hence the

Thus, referring once again to Table II., it will be seen
for example that, if the Breguet biplane receives 29 h.p.
in useful power from its propelling plant, the pilot, in
order to maintain horizontal flight, will have to manipulate
his elevator until the incidence of the planes is approxi-
mately 4°, which corresponds to the lift 0-040.

The speed, then, would only be about 80 km. per hour.

FLIGHT IN STILL AIR 31

Experience teaches the pilot to find the correct position
of the elevator to maintain horizontal flight. Should the
engine run irregularly, and if the aeroplane is to maintain
its horizontal flight, the elevator must be slightly actuated
in order to correct this disturbing influence.

Horizontal flight, therefore, implies a constant mainten-
ance of equilibrium, whence the designation equilibrator,
which is often applied to the elevator, derives full justifi-
cation.

But if the engine is running normally, the incidence, and
consequently the speed, of an aeroplane remain practically
constant, and these constitute its normal incidence and
speed.

Generally the engine is running at full power during
flight, and so in the ordinary course of events the normal
speed of an aeroplane is the highest it can attain.

But there is a growing tendency among pilots to
reserve a portion of the power which the engine is
capable of developing, and to throttle down in normal
flight. In this case the reserve of power available may
be saved for an emergency, and be used — the case will be
dealt with hereafter — for climbing rapidly, or to assume
a higher speed for the time being. In this case the
normal speed is, of course, no longer the highest possible
speed.

In the example already considered, the Breguet biplane
would fly at about 80 km. per hour, if it possessed useful
power amounting to 29 h.p.

But by throttling down the engine so that it normally
only produced a reduced useful power equivalent to 24 h.p.,
the normal speed of the machine, according to Table II.,
would only be 72 km. per hour (the normal incidence being
6|° and the lift 0-050).

The pilot would therefore have at his disposal a surplus
of power amounting to 5 h.p., which he could use, by,
opening the throttle, either for climbing or for temporarily
increasing his speed to 80 km. per hour.

32 FLIGHT WITHOUT FORMULA

Although, therefore, an aeroplane usually only flies at
one speed, which we call its normal speed, it can perfectly
well fly at other speeds, as was shown in Chapter I. But,
in order to obtain this result, it is essential that on each
occasion the engine should be made to develop the precise
amount of power required by the speed at which it is
desired to fly.

Speed variation can therefore only be achieved by
simultaneously varying the incidence and the power, or, in
practice, by operating the elevator and the throttle together.
This may be accomplished with greater or less ease
according to the type of motor in use, but certain pilots
practise it most cleverly and succeed in achieving a very
notable speed variation, which i? of great importance,
especially in the case of high-speed aeroplanes, at the
moment of alighting.

As has already been explained, the horizontal flight of an
aeroplane may be considered in the light of gliding flight
with the gliding angle artificially raised. From this point
of view it is possible to calculate in another way the power
required for horizontal flight.

For instance, if we know that an aeroplane of a given
weight, such as 600 kg., has, for a given incidence, a glid-
ing angle of 16 cm. per metre (approximately 1 in 6) at
which its speed is 22-3 m. per second, we conclude that
in 1 second it descended 0-16 x 22-3=3-58 m. Hence,
in order to overcome its descent and to preserve its hori-
zontal flight, it would be necessary to expend the useful
power required to raise a weight of 600 kg. to a height
of 3-58 m. in 1 second. Since 1 h.p. is the unit required
to raise a weight of 75 kg. to a height of 1 m. in 1 second,

the desired useful power = — —about 29 h.p. This,

as a matter of fact, is the amount given by Table II. for
the Breguet biplane which complies with the conditions
given.

In order to find the useful power required for the

FLIGHT IN STILL AIR 33

horizontal flight of an aeroplane flying at a given incidence,
and hence at a given speed, multiply the weight of the machine
by this speed and by the gliding angle corresponding to the
incidence, and divide by 75.

By a similar method one may easily calculate the useful
power required to convert horizontal flight into a climb at
any angle.

Thus, if the aeroplane already referred to had to climb,
always at the same speed of 22-3 m. per second, at an angle
of 5 cm. per metre (1 in 20), it would be necessary to expend
the additional power

0-05x600x22-3 ,

=about 9 h.p.

Of course, this expenditure of surplus power would be
greater the smaller the efficiency of the propeller, and
would be 12 h.p. for 75 per cent, efficiency, and 18 h.p.
for 50 per cent, efficiency.

Clearly, this method of making an aeroplane climb — by
increasing the motive power — can only be resorted to if
there is a surplus of power available, that is, if the engine
is not normally running at full power, which until now is
the exception.

For this reason, when, as is generally the case, the engine
is running at full power, climbing is effected in a much
simpler manner, which consists in increasing the angle of
incidence of the planes by means of the elevator.

Let us once more take our Breguet biplane which, with
motor working at full power, flies at a normal speed of
22-3 m. per second (80-3 km. per hour) at 4° incidence (or
a lift coefficient of 0-040). The useful power needed to
achieve this speed (see Table II.) is 29 h.p.

Assume that, by means of his elevator, the pilot increases
the angle of incidence to 10° (lift coefficient 0-060). Since
horizontal flight at this incidence, which must inevitably
reduce the speed to 18-2 m. per second or 65-6 km. per
hour, would only require 23 h.p., there will be an ex-

34 FLIGHT WITHOUT FORMULAE

cess of power amounting to 6 h.p.,* and the aeroplane
will rise.

The climbing angle can be calculated with great ease.
The method is just the converse of the one we have just
employed, and thus consists in dividing 6x75 (representing
the surplus power) by 600 x20 (weight multiplied by speed),
which gives an angle of 3*75 cm. per metre (1 in 27
about).

This climbing rate may not appear very great ; still, for
a speed of 18-2 m. per second, it corresponds to a climb
of 68 cm. per second=41 m. per minute=410 m. in 10
minutes, which is, at all events, appreciable.

The aeroplane, therefore, may be made to climb or to descend
by the operation of the elevator by the pilot.

More especially is the elevator used for starting. In
this case the elevator is placed in a position corresponding
to a very slight incidence of the main planes, so that these
offer very little resistance to forward motion when the
motor is started and the machine begins to run along the
ground. As soon as the rolling speed is deemed sufficient,
the elevator is moved to a considerable angle, which causes
the planes to assume a fairly high incidence, and the aero-
plane rises from the ground.

* This is not strictly correct, since, as will be seen hereafter, the
propeller efficiency varies to some extent with the speed of the aeroplane ;
still, we shall not make a grievous error in assuming that the efficiency
remains the same.

CHAPTER III

FLIGHT IN STILL AIR

POWER (concluded]

THE second chapter was mainly devoted to explaining how
one may calculate the useful power required for horizontal
flight, at the various angles of incidence and at the different
lift coefficients — in other words, at the various speeds of a
given aeroplane.

In addition, gliding flight has been briefly touched on,
and has served to show the precise manner in which the
power employed affects the speed of the aeroplane.

In the present chapter this discussion will be completed ;
it will be devoted to finding the best way of employing the
available power to obtain speed. Incidentally, we shall
have occasion to deal briefly with the limits of speed which
the aeroplane as we know it to-day seems capable of
attaining.

It has been shown that the flight of a given aeroplane
requires a minimum useful power, and that this is only
possible when the angle of incidence is that which we have
termed the economical angle.

The power would therefore be turned to the best account,
having regard merely to the sustentation of the aeroplane,
by making it fly normally at its economical angle.

But, on the other hand, this method is most defective
from the point of view of speed, for as fig. 6 (Chapter II.)
clearly shows, when the machine flies at its economical
angle, a very slight increase in power will increase the

36 FLIGHT WITHOUT FORMULA

speed to a considerable extent. Besides, the method in
question would be worthless from a practical point of view,
since it is evident that an aeroplane flying under these
conditions would be endangered by the slightest failure of
its engine.

Such, in fact, was the case with the first aeroplanes which
actually rose from the ground ; they flew " without a
margin," to use an expressive term. And even to-day the
same is true of machines whose motor is running badly :
in such a case the only thing to be done is to land as soon
as possible, since the aeroplane will scarcely respond to the
controls.

The other characteristic value of the angle of incidence
referred to in Chapter II., there called the optimum angle,
corresponds to the least value of the ratio between the
propeller-thrust and the weight of the aeroplane, or to its
equivalent — the best gliding angle.

For the best utilisation of the power in order to obtain
speed, which alone concerns us for the moment, there is a
distinct advantage attached to the use of the optimum
angle for the normal incidence of the machine ; Colonel
Renard, indeed, long ago pointed out that by using the
optimum angle for normal flight in preference to the
economical angle, one obtained 32 per cent, increase in speed
for an increase in power amounting to 13 per cent. only.

In any case, when the incidence is optimum the ratio
between the speed and the useful power required to obtain
it is largest. This is easily explained by reference to
Chapter II., which showed that the useful power required
for horizontal flight at a given incidence is proportional to
the speed multiplied by the gliding angle of the aeroplane
at the same incidence.

When the gliding angle is least (i.e. flattest), that is,
when the incidence is that of the optimum angle, the ratio
of power to speed is also smallest, and hence the ratio of
speed to maximum power.

It would therefore appear that by using the optimum

FLIGHT IN STILL AIR 37

angle as the normal incidence we would obtain the best
results from the point of view with which we are at present
concerned, which is that of the most profitable utilisation
of the power to produce speed. This, in fact, is generally
accepted as the truth, and in his scale model experiments
M. Eiffel always recorded this important value of the angle
of incidence, together with the corresponding flattest gliding
angle.

Nevertheless we are not prepared to accept as inevitably
true that the optimum angle is necessarily the most ad-
vantageous for flight, so far as the transmutation of power
into speed is concerned. This will now be shown by
approaching the question in a different manner, and by
finding the best conditions under which a given speed can
be attained.

The power required for flight is proportional, as has been
shown, to the propeller-thrust multiplied by the speed.
Hence, on comparing different aeroplanes flying at the same
speed, it will be found that the values of the power ex-
pended to maintain flight will have the same relation to one
another as the corresponding values of the propeller-thrust.

If we assume that the detrimental surface of each one
of these aeroplanes is identical, the head-resistance will be
the same in each case, since it is proportional to the detri-
mental surface multiplied by the square of the speed (which
is identical in every case).

It follows that the speed in question will be attained
most economically by the aeroplane whose planes exert
the least drag. Now, it was shown in Chapter II. that
the drag of the wings of an aeroplane is a fraction of the
weight of the machine equal to the ratio between the
drag coefficient and the lift coefficient corresponding to
the incidence at which flight is made.

If we assume, therefore, that the weight of each aero-
plane is identical, it follows that the best results are given
by that machine whose planes in normal flight have the
smallest drag-to-lift ratio.

38 FLIGHT WITHOUT FORMULA

Reference to the polar diagrams (Chapter I., figs. 1, 2, 3,
and 4) shows that the minimum drag-to-lift ratio occurs
at the angle of incidence corresponding to the point on
the curve where a straight line rotated about the centre
0-00 comes into contact with the curve. This angle of
incidence is beyond all question, for any aeroplane provided
with planes of the types under consideration, the most
profitable from our point of view ; this angle, in other
words, is that at which an aeroplane of given weight can
fly at a given speed for the least expenditure of power,
and this for any weight and speed. Hence this is the
angle at which an aeroplane possessing one of these wing
sections should always fly in theory. Accordingly, it may be
termed the be t angle of incidence, and the corresponding
lift coefficient the best lift coefficient.

The value of the best incidence only depends on the
wing section, but it is always smaller than the optimum
angle, which in its turn depends not only on the wing
section but also on the ratio of the detrimental surface to
the plane area.

A straight line rotated from the centre 0-00 in figs. 2, 3,
and 4 indicates that the best lift coefficients for M. Farman,
Breguet, and Bleriot XI. plane sections are respectively
0-017, 0-035, and 0-047, corresponding to the best angles
of incidence 1|°, 2£°, and 6°. These values can only be de-
termined with some difficulty, however, since the curves
are so nearly straight at these points that the rotating line
would come into contact with the curves for some distance
and not at one precise point alone.

On the other hand, it is evident that the drag-to-lift ratio
only varies very slightly for a series of angles of incidence,
the range depending on the particular plane section, so
that one is justified in saying that each type of wing pos-
sesses not only one best incidence and one best lift, but
several good incidences and good lifts.

Thus, for the Maurice Farman section, the good lifts lie
between 0-010 and 0-025 approximately, and the corre-

FLIGHT IN STILL AIR 39

spending good incidences extend from 1° to 4°, while the
drag-to-lift ratio between these limits remains practically
constant at 0-065.

For the Breguet wing, the good lifts are between 0-030
and 0-045, the good incidences between 3° and 6°, and the
drag-to-lift ratio remains about 0-08.

Lastly, for the Bleriot XI. the same values read as
follows : 0-030 and 0-055, 3° and 6°, and about 0-105.

Even at this point it becomes evident that the use of
.slightly cambered wings is the more suitable for flight
with a low lift coefficient, and that for a large lift a heavily
cambered wing is preferable.

If the optimum angle of an aeroplane, which depends,
as already shown, on the ratio between the detrimental
surface and the plane area, is included within the limits
of the good incidences, its use as the normal angle of in-
cidence remains as advantageous as that of any other
" good " incidence. But if it is not included,* flight at the
optimum angle would require, in theory at all events, a
greater expenditure of power than would be required under
similar conditions if flight took place at any of the good
incidences.

This shows that the optimum angle is not necessarily
that at which an aeroplane should fly normally in order to
use the power most advantageously.

To sum up : the normal speed should always correspond
to a " good " angle of incidence.

Should this not be the case in fact, it would be possible
to design an aeroplane which, for the same weight and
detrimental surface as the one under consideration, could
achieve an equal speed for a smaller expenditure of power.

A concrete example will render these considerations
clearer.

In Table II. (Chapter II.) there was set out the variation

* This would be possible more particularly in the case of aeroplanes
with very slightly cambered planes and small wing area and considerable
detrimental surface.

40 FLIGHT WITHOUT FORMULA

of the useful power required for the horizontal flight of a
Breguet aeroplane weighing 600 kg., with a plane area of
30 sq. m. and a detrimental surface of 1-20 sq. m., according
to its speed.

Let us assume that the useful power — 24 h. p. —developed
by the propeller makes the aeroplane fly normally at
0-050 lift, or at its optimum incidence. The speed will
then be 72 km. per hour or 20 m. per second. This lift
coefficient 0-050, be it noted, is slightly greater than the
highest of the good incidences peculiar to the Breguet
section.

Now let us take another aeroplane of the same type, also
weighing 600 kg. and with the same detrimental surface
of 1-20 sq. m., but with 40 sq. m. plane area, which should
still fly at the same speed of 20 m. per second.

The lift coefficient may be obtained (cf. Chapter I.) by
dividing the loading of the planes (15 kg.) by the square of
the speed in metres per second (400), which gives 0-0375.
Now this is one of the good lift coefficients of the Breguet
plane. In these conditions, therefore, the drag-to-lift ratio
will assume the constant value of about 0-08 common to all
good incidences.

It follows that the drag of the planes will be equal to
the weight, 600 kg. x 0-08=48 kg.

The head-resistance, on the other hand, will remain the
same as in the original aeroplane whose speed was 72 km.
per hour, since head-resistance is dependent simply on the
amount of detrimental surface and on the speed (neither of
which undergoes any change). The head-resistance, there-
fore (cf. Chapter II.), equals 38 kg.

The propeller-thrust, equal to the sum of head-resistance
and drag of the planes, will be 86 kg., and the useful power
required for flight =

Thrust (86) XqeedjjO^ ^

The figure thus obtained is less than the 24 h.p. of useful

FLIGHT IN STILL AIR 41

power required to make the aeroplane first considered fly
at 72 km. per hour.

Therefore, in theory at all events, the optimum angle is
not necessarily the most advantageous from the point of
view of the least expenditure of power to obtain speed.
But in practice the small saving in power would probably
be neutralised owing to the difficulty of constructing two
aeroplanes of the same type with a plane area of 30 and
40 sq. m. respectively without increasing the weight and
the detrimental surface of the latter. Hence the advantage
dealt with would appear to be purely a theoretical one in
the present case.

But this would not be so with an aeroplane whose normal
angle of incidence was smaller than the good incidences
belonging to its particular plane section. For instance, let
us assume that the propeller of the Breguet aeroplane
(vide Table II.) furnishes normally 68 useful h.p., which
would give the machine a speed of 113-6 km. per hour or
31-6 m. per second, at the lift 0-020, which is less than the
good lifts for this plane section.

Now take another Breguet aeroplane of the same weight
and detrimental surface, but with a plane area of only
20 sq. m. Calculating as before, it will be found that in
order to achieve a speed of 113-6 km. per hour, this machine
Avould have to fly with a lift of 0-030, which is one of the
good lifts, and that useful power amounting to only 60 h.p.
would be sufficient to effect the purpose. This time the
advantage of using a good incidence as the normal angle is
clearly apparent.

As a matter of fact, in practice the advantage would
probably be even more considerable, since a machine with
20 sq. m. plane area would probably be lighter and have
less detrimental surface than a 30 sq. m. machine.

Care should therefore be taken that the normal angle of
an aeroplane is included among the good incidences belonging
to its plane section, and, above all, that it is not smaller than
the good incidences.

42 PLIGHT WITHOUT FORMULA

This manner of considering good incidences and lifts
provides a solution of the following problem which was
referred to in Chapter I. :

Since there are only two means of increasing the speed
of an aeroplane — either by increasing the plane loading
or by reducing the lift coefficient — which of these is the more
economical ?

To begin with, the question will be examined from a
theoretical point of view, by assuming that the adoption of
either means will have the same effect in each case on the
weight and the detrimental surface, since the values of
these must be supposed to remain the same in the various
machines to enable our usual method of calculation to be
applied.

This being so, it will be readily seen that as long as the
normal lift remains one of the good lifts, both means of
increasing the speed are equivalent as far as the expenditure
of useful power is concerned.

On the one hand, since the drag-to-lift ratio retains
approximately the same value for all good lifts, the drag
of the planes will remain for every angle of incidence a
constant fraction of the weight, which is assumed to be in-
variable. On the other hand, at the speed it is desired to
attain, the head-resistance, proportional to the detrimental
surface, which is also assumed to be invariable, will remain
the same in both cases. Consequently, the propeller-thrust,
equal to the sum of the two resistances (drag of the planes
4-head-resistance), and hence the useful power, will retain
the same value by whichever of the two methods the increase
in speed has been obtained.

But if the lift had already been reduced to the smallest
of the good lift values, and it was still desired to increase
the speed, the most profitable manner of doing this would
be to increase the loading by reducing the plane area. So
much for the theoretical aspect of the problem.

Purely practical considerations strengthen these theor-
etical conclusions, in so far as they clearly prove the ad-

FLIGHT IN STILL AIR 43

vantage of increasing the speed by the reduction of plane
area, even where the lift remains one of the good lift
values.

Indeed, in practice the two methods are no longer equiva-
lent in the latter case, since, as already mentioned, the
reduction of the wing area is usually accompanied by a
decrease in the weight and detrimental surface.

Generally speaking, it is therefore preferable to take the
highest rather than the lowest of the good lifts as the
normal angle of incidence, and this conclusion tallies,
moreover, with that arising from the danger of flying
at a very low lift. Finally, the normal angle would thus
remain in the neighbourhood of the optimum angle,
which is an excellent point so far as a flat gliding angle
is concerned.*

Obviously, the advantage of the method of increasing
the speed by reducing the plane area over that consisting
in reducing the lift becomes greater still in the case where
the latter method, if applied, would lead to the lift being
less than any of the good lift values.

The disadvantage of greatly reducing the plane area
to obtain fast machines is the heavy loading which it
entails and the lessening of the gliding qualities. The best
practical solution of the whole problem would therefore
appear to consist in a judicious compromise between these
two methods.

As usual, a concrete example will aid the explanation
given above.

Let the Breguet aeroplane already referred to be supposed
to fly at a speed of 92-8 km. per hour with a lift of 0-030,
which is the lowest of its good lift values. Table II. shows
that this would require 38 h.p.

Another machine of the same type, and having the same
weight and detrimental surface, but with an area of only
20 sq. m. (instead of 30), in order to attain the same speed

* Chapter X. will show that this conclusion is strengthened still
further by the effect of wind on the aeroplane.

44 FLIGHT WITHOUT FORMULA

would have to fly at 0-040 lift, which is also one of the
good lift values.

The necessary calculations would show that the latter
machine, like the former, would also require 38 h.p. This is
readily explicable on the score that the drag of the planes
is 0-08 of the weight, or 48 kg., while the head-resistance
also remains constant and equal to 64 kg. (Table II.).

In theory, therefore, there is nothing to choose between
either solution. But in practice the latter is preferable,
since the 20 sq. m. machine would in all likelihood be lighter
and possess less detrimental surface.

But if a speed of 113-6 km. per hour were to be attained,
the 20 sq. m. aeroplane has a distinct advantage both in
theory, and even more in practice, for the machine with
30 sq. m. area would have to fly at 0-020 lift, which is
lower than the good lift values belonging to the Breguet
plane section, which would, as already shown, require
useful power amounting to 68 h.p., whereas 60 h.p. would
suffice to maintain the smaller machine in flight at the
same speed.

We have already set forth the good lift values belonging
to the Maurice Farman, Breguet, and Bleriot XI. plane
sections, and the corresponding values of the drag-to-lift
ratio or, its equivalent, the ratio of the drag of the planes
to the weight of the machine. .

Reference to these values has already shown that slightly
cambered planes are undoubtedly more economical for low
lift values, which are necessary for the attainment of high
speeds, especially in the case of lightly loaded planes, as in
some biplanes.

But the good lift values of very flat planes are usually
very low — from 0-010 to 0-025 in the case of the Maurice
Farman — which greatly restricts the use of these values,
since, as already stated, it is doubtful whether hitherto an
aeroplane has flown at a lower lift value than 0-020.

The advantages and disadvantages of these three wing
sections, from the point of view at issue, will be more readily

FLIGHT IN STILL AIR

?RIOT
0^

n°XI

EGUE

TO^

/'~

M.I

'ARM*

I/V

1 o1
• 1
9

, i

4 !

O.Q8

0.07

0.08

seen by plotting their polar curves in one diagram, as
shown in fig. 11.

The Breguet and Maurice Farman curves intersect at a
point corresponding to the
lift value 0-030, whence we
may conclude that for all lift
values lower than this, the
Maurice Farman section is
the better,* but for all lift
values higher than 0-032
(which at present are more
usual), the Breguet wing
has a distinct advantage.
In the same way, the
Maurice Farman is better
than the Bleriot XI. for
lift values below 0-042,
whereas the latter is better
for all higher lift values.

Finally, the Bleriot XI.
only becomes superior to
the Breguet for lift values
in excess of 0-065, which
are very high indeed, and
little used owing to the
fact that they correspond
to angles in the neighbour-
hood of the economical
angle.

To apply these various
considerations, we will now
proceed to fix the best con-
ditions in which to obtain

A 0.04

0.03

0.02

0.01

0.02
Drag.

0.01

o'uo

o.oq

FIG. 11

a speed of 160 km. per
hour or about 44-5 m. per second, which appears to be
the highest speed which it seems at present possible to
* Since it has a smaller drag for the same lift.

46 FLIGHT WITHOUT FORMULA

reach,* that is, by assuming it to be possible to have a
loading of 40 kg. per sq. m. of surface and to fly at a lift-
value of 0-020.

In laying down this limit to the speed of flight we also
stated our belief that, in order to enable it to be attained,
engines developing from 120 to 130 effective h.p. would
have to be employed.

This opinion was founded on the results of M. Eiffel's
experiments, from which it was concluded that an aeroplane
to attain this speed would have to possess a detrimental
surface of no more than 0-75 sq. m.

Now, the last two Aeronautical Salons, those of 1911
and 1912, have shown a very clearly marked tendency
among constructors to reduce all passive resistance to the
lowest possible point, especially in high-speed machines,
and it would appear that in this direction considerable
progress has been and is being made.

One machine in particular, the Paulhan-Tatin " Torpille,"
specially designed with this point in view, is worthy of
notice.

Its designer, the late M. Tatin, estimated the detrimental
surface of this aeroplane at no more than 0-26 sq. m., and
its resistance must in fact have been very low, since it had
the fair-shaped lines of a bird, every part of the structure
capable of setting up resistance being enclosed in a shell-like
hull from which only the landing wheels, reduced to the
utmost verge of simplicity, projected.

Taking into account the slightly less favourable figures
obtained by M. Eiffel from experiments with a scale model,
the detrimental surface of the " Torpille " may be estimated
at 0-30 sq. m.

According to information given by M. Tatin himself, the
weight of this monoplane was 450 kg., and its plane area
12-5 sq. m.

* It should, however, be remembered that this limit has actual^
been exceeded, with a loading of 44 kg. per sq. m. and a lift value of
slightly less than 0*020. See also Translator's note on p. 12.

FLIGHT IN STILL AIR 47

Let us assume that the planes, which were only very
slightly cambered, were about equivalent to those of the
Maurice Farman, and that they flew at a good lift coefficient.
In that case the drag of the planes would be equal to 0-065
of the weight of the machine, or to 29-5 kg.

On the other hand, at the speed of 44-5 m. a second, the
head-resistance =

n ffi • Detrimental Square of

Coefficient. ^^ ^ s^

0-08 x 0-3 X 1980 - 47-5 kg.

The propeller-thrust, consequently, the sum of both
resistances, would =17 kg.

The useful power required would thus=

77x44-5 ,

=about 45 h.p.

Propeller efficiency in this case must have been exception-
ally high (as will be seen hereafter), and was probably in
the region of 80 per cent.

The engine -power required to give the " Aero-Torpille " a

A£*

speed of 160 km. per hour must therefore have been — =

0*8

57 h.p., or approximately 60 h.p.

M. Tatin considered that he could obtain the same result
with even less motive-power, and that some 45 h.p. would
suffice. If this proves to be the case, the detrimental
surface of the aeroplane would have to be less than 0-30
sq. m. and the propeller efficiency even higher than 80 per
cent., or else — and this was M. Tatin's own opinion — the
coefficients derived from experiments with small scale
models must be increased for full-size machines, their
value possibly depending in some degree on the speed.*

* No proof, as a matter of fact, was possible owing to the short life of
the machine. But the results obtained from other machines in which
stream-lining had been carried out to an unusual degree, such as the
Deperdussin " monocoque " — which, with an engine of 85-90 effective
h.p., only achieved 163 km. per hour — would appear to show that the

48 FLIGHT WITHOUT FORMULA

It should also be noted that, in order to attain 160 km.
per hour, the Tatin " Torpille " would have to fly at a lift
coefficient equal to

36 (loading) =0-018

1980 (square of the speed)

Perhaps it will seem strange that simply by estimating
the value of the detrimental surface at 0-30 instead of the
previous estimate of 0-75, the motive power required for
flight at 160 km. per hour should have been reduced by
one-half. Yet there is no need for surprise ; for if the
method for calculating the useful power necessary for
horizontal flight (set forth in Chapter II., and since applied
more than once) is carefully examined, it becomes evident
that, whereas that portion of the power required only for
lifting remains proportional to the speed, the remaining
portion, used to overcome all passive resistance, is propor-
tional to the cube of the speed.

For this reason it is of such great importance to cut down
the detrimental surface hi designing a high-speed machine.

Thus, in the present case, of the 46 h.p. available, only
18 h.p. are required to lift the machine. The remain-
ing 28 h.p., therefore, are necessary to overcome passive
resistance.

Had the detrimental surface been 0-75 sq. m. instead of
0-30, the useful power absorbed in overcoming passive
resistance would have been

Q.75x28=7() h mgtead 2g
0-30

To complete our examination of the high-speed aeroplane,
Table IV. has been drawn up, and includes the values of the
useful power required on the one hand for the flight of a
Maurice Farman plane at a good incidence, and weighing

estimate of 0'30 sq. m. for the detrimental surface was too low, a con-
clusion supported by M. Eiffel's experiments.

It is doubtful whether an aeroplane has yet been built with a detri-
mental surface of much less than half a square metre.

FLIGHT IN STILL AIR

1 ton (metric), and on the other for driving through
air a detrimental surface of 1 sq. m. at speeds from
to 200 km. per hour.

TABLE IV.

the
150

Speed.

lllL

II. J^s

|3*g*-i

|||.si

Km.

Metres

Drag of
Planes (kg.) ;
per aeroplane
ton.

-sll^l

1151II

fsflll

Ijjn

per hr.

per see.

"* £ ^ >> "

3 f *" ^oo g

^ f, 2 -S ^

g * ^*>

(2«S ^6

}S «se^ 8

150

41-6

138

160

44-4^

•slsf

157

170

47-2

l?|ol

178

112

180
190

50-0
52-8

43
46

200
232

132
157

200

55-6,

-** K 5

248

184

•^ o

According to this Table, an aeroplane weighing 500 kg.,
and possessing, as we supposed in the case of the Tatin
" Torpille," a detrimental surface of 0-30 sq. m., would re-
quire a useful power of about 80 h.p. to attain a speed of
200 km. per hour. This high speed could therefore be
achieved with a power-plant consisting of a 100-h.p. motor
and a propeller of 80 per cent, efficiency. It could only be
obtained — just as the " Torpille " could only achieve 160
km. per hour at a lift coefficient of 0-018 — with a plane
loading of about 56 kg. per sq. m. Consequently, the

area of the planes would be only -=^- = 9 sq. m.

If the theoretical qualities of design of machines of the
" Torpille " type are borne out by practice * our present

* But, according to what has already been said, this does not seem to
be the case. Hence, a speed of 200 km. per hour is not likely to be

50 FLIGHT WITHOUT FORMULA

motors would appear to be sufficient to give them a speed
of 200 km. per hour. But this would necessitate a very
heavy loading and a lift coefficient much lower than any
hitherto employed — a proceeding which, as we have seen, is
not without danger. Moreover, one cannot but be uneasy
at the thought of a machine weighing perhaps 500 or 600
kg. alighting at this speed.

This, beyond all manner of doubt, is the main obstacle
which the high-speed aeroplane will have to overcome, and
this it can only do by possessing speed variation to an
exceptional degree. We will return to this aspect of the
matter subsequently.

To-day an aeroplane, weighing with full load a certain
weight and equipped with an engine giving a certain power,
in practice flies horizontally at a given speed.

These three factors, weight, speed, and power, are always
met with whatever the vehicle of locomotion under con-
sideration, and their combination enables us to determine
as the most efficient from a mechanical point of view that
vehicle or machine which requires the least power to attain,
for a constant weight, the same speed.

Hence, what we may term the, mechanical efficiency
of an aeroplane may be measured through its weight
multiplied by its normal speed and divided by the motive-
power.

If the speed is given in metres per second and the power
in h.p., this quotient must be divided by 75.

RULE. — The mechanical efficiency of an aeroplane is
obtained by dividing its weight multiplied by its normal
speed (in metres per second) by 75 times the power, or, what
is the same thing, by dividing by 270 times the power the
product of the weight multiplied by the speed in kilometres
per hour.

EXAMPLE. — An aeroplane weighing 950 kg., and driven

attained with a 100-h.p. motor. Whether an engine developing 140 h.p.
or more will succeed in this can only be shown by the future, and perhaps
at no distant date. See footnote, p. 12.

FLIGHT IN STILL AIR 51

by a, lOQ-h.p. engine, flies at a normal speed of 117 km. per
hour. What is its mechanical efficiency ?

950x117

Reference to what has already been said will show that
mechanical efficiency is also expressed by the propeller
efficiency divided by the gliding angle corresponding to
normal incidence. This is due to the fact that, firstly, the
useful power required for horizontal flight is the 75th part
of the weight multiplied by the speed and the normal
gliding angle, and, secondly, because the motive power is
obtained by dividing the useful power by the propeller
efficiency. Accordingly, a machine with a propeller efficiency
of 70 per cent., and with a normal gliding angle of 0-17,

would have a mechanical efficiency—— — =4-12.

This conception of mechanical efficiency enables us to
judge an aeroplane as a whole from its practical flying
performances without having recourse to the propeller
efficiency and the normal gliding angle, which are difficult
to measure with any accuracy.

Even yesterday a machine possessing mechanical efficiency
superior to 4 was still, aerodynamically considered, an
excellent aeroplane. But the progress manifest in the
last Salon entitles us, and with confidence, to be more
exacting in the future.

Hence, the average mechanical efficiency of the ordinary
run of aeroplanes enables us in some measure to fix definite
periods in the history of aviation. In 1910, for instance,
the mean mechanical efficiency was roughly 3-33, on which
we based the statement contained in a previous work that,
in practice, 1 h.p. transports 250 kg. in the case of an average
aeroplane at 1 m. per second.

This rule, which obviously only yielded approximate
results, could be applied both quickly and easily, and enabled
one, for instance, to form a very fair idea of the results

52 FLIGHT WITHOUT FORMULA

that would be attained in the Military Trials of 1911. In
fact, according to the rules of this competition, the aero-
planes would have to weigh on an average 900 kg. To

give them a speed of 70 km. per hour or — - m. per second,

3'6

c • i A • 90° v 70

for instance, the rule quoted gives — — x^-^-

250 o'b

But 250 X 3-6 remains the denominator whatever the speed
it is desired to attain, and is exactly equal to 900, the weight
of the aeroplane. From this, one deduced that in this case
the power required in h.p. was equivalent to the speed in
kilometres per hour : —

70 km. per hour 70 h.p.

80 „ ..... 80 h.p.
100 ,, 100 h.p.

If, on the other hand, certain machines during these trials,
driven by engines developing less than 100 effective h.p.,
flew at over 100 km. per hour, this was due simply to their
mechanical efficiency being better than the 3-33 which
obtained in 1910, and was already too low for 1911.

At the present day, therefore, accepting 4 as the average
mechanical efficiency, the practical rule given above should
be modified as follows : —

RULE. — 1 h.p. transports 300 kg. of an average aeroplane
at 1 m. per second.

CHAPTER IV

FLIGHT IN STILL AIR
THE POWER-PLANT

BOTH this chapter and the next will be devoted to the
power-plant of the aeroplane as it is in use at the present
time. This will entail an even closer consideration of the
part played by the motive-power in horizontal and oblique
flight, and will finally lead to several important conclusions
concerning the variable-speed aeroplane and the solution of
the problem of speed variation.

The power-plant of an aeroplane consists in every case
of an internal combustion motor and one or more propellers.
Since the present work is mainly theoretical, no description
of aviation motors will be attempted, and only those of
their properties will be dealt with which affect the working
of the propeller.

Besides, the motor works on principles which are beyond
the realm of aerodynamics, so that from our point of view
its study has only a minor interest. It forms, it is true, an
essential auxiliary of the aeroplane, but only an auxiliary.
If it is not yet perfectly reliable, there is no doubt that it
will be in a few years, and this quite independently of any
progress in the science of aerodynamics.

Deeply interesting, on the other hand, are the problems
relating to the aeroplane itself, or to that mysterious
contrivance which, as it were, screws itself into the air
and transmutes into thrust the power developed by the
engine.

54 FLIGHT WITHOUT FORMULA

The power developed by an internal combustion engine
varies with the number of revolutions at which the resist-
ance it encounters enables it to turn. There is a generally
recognised ratio between the power developed and the speed
of revolution.

Thus, if a motor, normally developing 50 h.p. at 1200
revolutions per minute, only turns at 960 revolutions per

50 X 960
minute, it will develop no more than =40 h.p.

The rule, however, is not wholly accurate, and the
variation of the power developed by a motor with the
number of revolutions per minute is more accurately shown
in the curve in fig. 12. It should be clearly understood
that the curve only relates to a motor with the throttle
fully open, and where the variation in its speed of rotation
is only due to the resistance it has to overcome.

For the speed of rotation may be reduced in another
manner — by shutting off a portion of the petrol mixture by
means of the throttle. The engine then runs "throttled
down," which is the usual case with a motor car.

In such a case, if the petrol supply is constant, the curve
in fig. 12 grows flatter, with its crest corresponding to a
lower speed of rotation the more the throttle is closed and
the explosive mixture reduced.

Fig. 13 shows a series of curves which were prepared at
my request by the managing director of the Gnome Engine
Company ; these represent the variation in power with the
speed of rotation of a 50-h.p. engine, normally running at
1200 revolutions per minute, with the throttle closed to a
varying extent.

In practice, it is easier to throttle down certain engines
than others ; with some it is constantly done, with others
it is more difficult.

Even to-day the working of a propeller remains one of
the most difficult problems awaiting solution in the whole
range of aerodynamics, and the motion, possibly whirling,
of the air molecules as they are drawn into the revolving

FLIGHT IN STILL AIR

Horse-Power.

& 3 3..

FLIGHT WITHOUT FORMULA

Horse-Power.

.5 $ § * § J? 3 3 3 ^ Lja__2

(

FLIGHT IN STILL AIR 57

propeller has never yet been explained in a manner satis-
factory to the dictates of science.

All said and done, the rough method of likening a propeller
to a screw seems the most likely to explain the results
obtained from experiments with propellers.

The pitch of a screw is the distance it advances in one
revolution in a solid body. The term may be applied in a
similar capacity to a propeller. The pitch of a propeller,
therefore, is the distance it would travel forwards during
one revolution if it could be made to penetrate a solid body.
But a propeller obtains its thrust from the reaction of an
elusive tenuous fluid. Clearly, therefore, it will not travel
forward as great a distance for each revolution as it would
if screwing itself into a solid.

The distance of its forward travel is consequently always
smaller than the pitch, and the difference is known as the
slip. But, contrary to an opinion which is often held, this
slip should not be as small as possible, or even be altogether
eliminated, for the propeller to work under the best con-
ditions.

Without attempting to lay down precisely the phenomena
produced in the working of this mysterious contrivance, we
may readily assume that at every point the blade meets the
air, or " bites " into it, at a certain angle depending, among
other things, on the speed of rotation and of forward travel
of the blade and of the distance of each point from the
axis.

Just as the plane of an aeroplane meeting the air along
its chord would produce no lift, so a propeller travelling
forward at its pitch speed — that is, without any slip —
would meet the air at each point of the blades at no angle
of incidence, and consequently would produce no thrust.

The slip and angle of incidence are clearly connected
together, and it will be easily understood that a given
propeller running at a given number of revolutions will
have a best slip, and hence a lest forward travel, just as
a given plane has a best angle of incidence.

58 FLIGHT WITHOUT FORMULA

When the propeller rotates without moving forward
through the air, as when an aeroplane is held stationary on
the ground, it simply acts as a ventilator, throwing the air
backwards, and exerts a thrust on the machine to which
it is attached. But it produces no useful power, for in
mechanics power always connotes motion.

But if the machine were not fixed, as in the case of an
aeroplane, and could yield to the thrust of the propeller,
it would be driven forward at a certain speed, and the
product of this speed multiplied by the thrust and divided
by 75 represents the useful power produced by the pro-
peller.

On the other hand, in order to make the propeller rotate
it must be acted upon by a certain amount of motive power.
The relation between the useful power actually developed
and the motive power expended is the efficiency of the
propeller.

But the conditions under which this is accomplished
vary, firstly, with the number of revolutions per minute
at which the propeller turns, and secondly, with the speed
of its forward travel, so that it will be readily understood
that the efficiency of a propeller may vary according to the
conditions under which it is used.

Experiments lately conducted — notably by Major
Dorand at the military laboratory of Chalais-Meudoii and
by M. Eiffel — have shown that the efficiency remains
approximately constant so long as the ratio of the forward
speed to its speed of revolution, i.e. the forward travel per
revolution, remains constant.

For instance, if a propeller is travelling forward at 15
m. per second and revolving at 10 revolutions per second,
its efficiency is the same as if it travelled forward at 30
m. per second and revolved at 20 revolutions per second,
since in both cases its forward travel per revolution is
1-50 m.

But the propeller efficiency varies with the amount of
its forward travel per revolution.

FLIGHT IN STILL AIR 59

Hence, when the propeller revolves attached to a
stationary point, as during a bench test, so that its forward
travel is zero, its efficiency is also zero, for the only
effect of the motive power expended to rotate the propeller
is to produce a thrust, which in this instance is exerted
upon an immovable body, and therefore is wasted so far
as the production of useful power is concerned.

Similarly, when the forward travel of the propeller per
revolution is equal to the pitch, and hence when there is
no slip, it screws itself into the air like a screw into a
solid ; the blades have no angle of incidence, and therefore
produce no thrust.*

Between the two values of the forward travel per
revolution at which the thrust disappears, there is a value
corresponding on the other hand to maximum thrust.
This has already been pointed out, and has been termed
the best forward travel per revolution.

This shows that the thrust of one and the same propeller
may vary from zero to a maximum value obtained with a
certain definite value of the forward travel. The variation
of the thrust with the forward travel per revolution may
be plotted in a curve. A single curve may be drawn to
show this variation for a whole family of propellers, geometri-
cally similar and only differing one from the other by their
diameter.

Experiments, in fact, have shown that such propellers
had approximately the same thrust when their forward
travel per revolution remained proportional to their
diameter.

Thus two propellers of similar type, with diameters
measuring respectively 2 and 3m., would give the same
thrust if the former travelled 1-2 m. per revolution and

* This could never take place if the vehicle to which the propeller
was attached derived its speed solely from the propeller ; it could only
occur in practice if motive power from some outside source imparted
to the vehicle a greater speed than that obtained from the propeller-
thrust alone.

FLIGHT WITHOUT FORMULA

the latter 1-8 m., since the ratio of forward travel to
diameter =0-60.

This has led M. Eiffel to adopt as his variable quantity
not the forward travel per revolution, but the ratio of this
advance to the diameter, which ratio may be termed
reduced forward travel or advance.

Fig. 14, based on his researches, shows the variation in
thrust of a family of propellers when the reduced advance
assumes a series of gradually increasing values.*

The maximum thrust efficiency (about 65 per cent, in
this case) corresponds to a reduced advance value of 0-6.

Reduced Advance.
FIG. 14.

Hence a propeller of the type under consideration, with a
diameter of 2-5 m., in order to give its highest thrust,
would have to have a forward travel of 2-5 xO'6=l-2 m.
Consequently, if in normal flight it turned at 1200
revolutions per minute, or 20 revolutions per second, the
machine it propelled ought to fly at 1-20x20=24 m. per
second.

For all propellers belonging to the same family there
exists, therefore, a definite reduced advance which is more

* Actually, M. Eiffel found that for the same value of the reduced
advance the thrust was not absolutely constant, but rather that it
tended to grow as the number of revolutions of the propeller increased.
Accordingly, he drew up a series of curves, but these approximate very
closely one to another.

FLIGHT IN STILL AIR

favourable than any other, and may thence be termed the
best reduced advance, which enables any of these propellers
to produce their maximum thrust.

It has been shown that all geometrically similar pro-
pellers— in other words, belonging to the same family —
give approximately the same maximum thrust efficiency.

But when the shape of the propeller is changed, this
maximum thrust value also varies.

It depends more especially on the ratio between the
pitch of the propeller and its diameter, which is known as
the pitch ratio.

But, as the value of the highest thrust varies with the
pitch ratio, so does that of the best reduced advance corre-
sponding to this highest thrust.

In the following Table V., based on Commandant Dorand's
researches at the military laboratory of Chalais-Meudon
with a particular type of propeller, are shown the values
of the maximum thrust and the best reduced advance
corresponding to propellers of varying pitch ratio.

TABLE V.

Pitch ratio .

0-5

0-6

0-7

0-8

0-9

1-0

I'M

Maximum thrust efficiency

0-45

0'53

0'61

0-70

0-76

0-80

0-84

Best reduced advance

0"29

0-38

0-47

0-55

0-63

072

0-84

EXAMPLE. — A propeller of the Chalais-Meudon type with
2-5 m. diameter and 2 m. pitch turns at 1200 revolutions
per minute.

1. What is the value of its highest thrust efficiency ?

2. What should be the speed of the aeroplane it drives in
order to obtain this highest thrust ?

The pitch ratio is — ~=0-8.
Table V. immediately solves the first question : the

62 FLIGHT WITHOUT FORMULA

highest thrust efficiency is 0-7. Further, this table shows
that to obtain this thrust the reduced advance should
=0-55. In other words, the speed of the machine divided
by 50 (the number of revolutions per second, 50 x diameter,
2-5) should =0-55.

Hence the speed =0-55 x 50=27-5 m. per second, or
99 km. per hour.

Again, Table V. proves, according to Commandant
Dorand's experiments, that even at the present time it is
possible to produce propellers giving the excellent efficiency
of 84 per cent, under the most favourable running conditions,
but only if the pitch ratio is greater than unity — that is,
when the pitch is equal to or greater than the diameter.

It is further clear that, since the best reduced advance
increases with the pitch ratio, the speed at which the
machine should fly for the propeller (turning at a constant
number of revolutions per minute) to give maximum
efficiency is the higher the greater the pitch ratio. This is
why propellers with a high pitch ratio, or the equivalent,
a high maximum thrust, are more especially adaptable
for high-speed aeroplanes. At the same time, they are
equally efficient when fitted to slower machines, provided
that the revolutions per minute are reduced by means of
gearing.

These truths are only slowly gaining acceptance to-day —
although the writer advocated them ardently long since, —
and this notwithstanding the fact that the astonishing
dynamic efficiency of the first motor-driven aeroplane
which in 1903 enabled the Wrights, to their enduring
glory, to make the first flight in history, was largely due to
the use of propellers with a very high pitch ratio, that is,
of high efficiency, excellently well adapted to the relatively
low speed of the machine by the employment of a good
gearing system.

The only thing that seemed to have been taught by this
fine example was the use of large diameter propellers.

This soon became the fashion. But, instead of gearing

FLIGHT IN STILL AIR 63

down these large propellers, as the Wrights cleverly did,
they were usually driven direct by the motor, and so that
the latter could revolve at its normal number of revolutions
the pitch had perforce to be reduced.

As the pitch decreased, so the maximum efficiency and
the best reduced advance — that is, the most suitable flying
speed — fell off, while at the same time the development of
the monoplane actually led to a considerable increase in
flying speed.

The result was that fast machines had to be equipped
with propellers of very low efficiency which, even so, they
were unable to attain, as the flying speed of the aeroplane
was too high for them. At most these propellers might
have done for a dirigible, but they would have been poor
even at that.

Fortunately, a few constructors were aware of these
facts, and to this alone we may ascribe the extraordinary
superiority shown towards the end of 1910 by a few types
of aeroplanes, among which we may name, without fear
of being accused of bias, those of M. Breguet and the late
M. Nieuport.

But, since then, progress has been on the right lines,
and those who visited the last three Aero shows must have
been struck with the general decrease in propeller diameter,
which has been accompanied by an increase in efficiency
and adaptability to the aeroplanes of to-day.

To take but one final example : the fast Paulhan-Tatin
" Torpille," already referred to, had a pitch ratio greater
than unity. For this reason its efficiency was estimated in
the neighbourhood of 80 per cent.

The foregoing considerations may be summed up as
follows : —

1. The same propeller gives an efficiency varying accord-
ing to the conditions in which it is run, depending on its
forward travel per revolution.

2. Each propeller has a speed of forward travel or advance
enabling it to produce its highest efficiency.

64 FLIGHT WITHOUT FORMULA

3. For propellers of identical type but different diameters
the various speeds of forward travel corresponding to the
same thrust are proportional to the diameters, whence
arises the factor of reduced advance, which, in other words,
is the ratio between the forward travel per revolution and
the diameter.

4. The maximum efficiency of a propeller and its best
reduced advance depend on its shape, and more especially
on its pitch ratio.

Hitherto the propeller has been considered as a separate
entity, but in practice it works in conjunction with a
petrol motor, whether by direct drive or gearing.

But the engine and propeller together constitute the
power-plant, and this new entity possesses, by reason of the
peculiar nature of the petrol motor, certain properties
which, differing materially from those of the propeller by
itself, must therefore be considered separately.

First, we will deal with the case of a propeller driven
direct off the engine.

Let us assume that on a truck forming part of a railway
tram there has been installed a propelling plant (wholly
insufficient to move the tram) consisting of a 50-h.p. motor
running at 1200 revolutions per minute, and of a propeller,
while a dynamometer enables the thrust to be constantly
measured and a revolution indicator shows the revolutions
per minute.

The tram being stationary, the motor is started.

The revolutions will then attain a certain number, 950
revolutions per minute for instance, at which the power
developed by the motor is exactly absorbed by the propeller.
The latter will exert a certain thrust upon the train (which,
of course, remains stationary), indicated by the dynamo-
meter and amounting to, say, 150 kg.

The power developed by the motor at 950 revolutions per
minute is shown by the power curve of the motor, which
we will assume to be that shown in fig. 12. This would
give about 43 h.p. at 950 revolutions per minute.

FLIGHT IN STILL AIR 65

The useful power, on the other hand, is zero, since no
movement has taken place.

Now let the train be started and run at, say, 10 km. per
hour or 5 m. per second, the motor still continuing to run.

The revolutions per minute of the propeller would
immediately increase, and finally amount to, say, 1010
revolutions per minute.

The power developed by the motor would therefore have
increased and would now amount, according to fig. 12, to
45-5 h.p.

But at the same time the dynamometer would show a
smaller thrust — about 130 kg.

But this thrust would, though in only a slight degree,
have assisted to propel the train forward and the useful

power produced by the propeller would be =8*7 h.p.

The acceleration in rotary velocity and the decrease in
thrust which are thus experienced are to be explained on
the score that the blades, travelling forward at the same
time that they revolve, meet the air at a smaller angle
than when revolving while the propeller is stationary.
In these conditions, therefore, the propeller turns at
a greater number of revolutions, though the thrust falls
off.

If the speed of the train were successively increased to
10, 15 and 20 m. per second, the following values would be
established each time : —

The normal number of revolutions of the power-plant ;

The corresponding power developed by the motor ;

The useful power produced by the propeller.

We could then plot curves similar to that shown in fig. 15,
giving for every speed of the train the corresponding
motive power (shown in the upper curve) and the useful
power (lower curve). The dotted lines and numbers give
the number of revolutions.

The lower curve representing the variation in the useful
power produced by the propeller according to the forward

FLIGHT WITHOUT FORMULA

speed of travel is of capital importance, and will hereafter
be referred to as the power-plant curve.

Usually the highest points of the two curves, L and M,
do not correspond. This simply means that generally, and
unless precautions have been taken to avoid this, the pro-
peller gives its maximum thrust, and accordingly has its
best reduced advance, at a forward speed which does not

950
40

k-d of fliffkt (in

Q i

FIG. 15.

enable the motor to turn at its normal number of revolu-
tions, 1200 in the present case, and consequently to develop
its full power of 50 h.p.

It is even now apparent, therefore, that one cannot mount
any propeller on any motor, if direct-driven, and that there
exists, apart altogether from the machine which they drive,
a mutual relation between the two parts constituting the
power-plant, which we will term the proper adaptation of
the propeller to the motor.

FLIGHT IN STILL AIR 67

Its characteristic feature is that the highest points in the
two curves representing the values of the motive power
and the useful power at different speeds of flights lie in a
perpendicular line (see fig. 16).

The highest thrust efficiency is then obtained from the
propeller at such a speed that the motor can also develop
its maximum power.

Spekd of fh$k

10 (5

FIG. 16.

The expression maximum power-plant efficiency will be
used to denote the ratio of maximum useful power Mm
(see fig. 15) developed at the maximum power LJ of which
the motor is capable (50 h.p. in the case under consideration).

The maximum power-plant efficiency, it is clear, corre-
sponds to a certain definite speed of flight Om. This may
be termed the best speed suited to the power-plant.

If the adaptation of the propeller to the motor is good
(as in the case of fig. 16), the maximum power-plant
efficiency is the highest that can be obtained by mounting

68 FLIGHT WITHOUT FORMULA

direct-driven propellers belonging to one and the same
family and of different diameters on the motor.

Hence there is only one propeller in any family or series
of propellers which is well adapted to a given motor.

We already know that in a family of propellers the
characteristic feature is a common value of the pitch ratio
— supposing, naturally, that the propellers are identical in
other respects. The conclusion set down above can there-
fore also be expressed as follows : —

There can be only one propeller of given pitch ratio that
is well adapted to a given motor. The diameter of the pro-
peller depends on the pitch ratio, and vice versa.

Propellers well adapted to a given motor consequently
form a single series such that each value of the diameter
corresponds to a single value of the pitch, and vice versa.

According to the results of Commandant Dorand's experi-
ments with the type of propellers which he employed, the
series of propellers properly adapted to a 50 h.p. motor
turning at 1200 revolutions per minute can be set out as
in Table VI., which also gives the best speed suited to the
power-plant in each case, and the maximum useful powers
developed obtained by multiplying the power of the motor,
50 h.p., by the maximum efficiency as given in Table V.

To summarise :

1. The useful power developed by a given power-plant
varies with the speed of the aeroplane on which it is mounted.
The variation can be shown by a curve termed the char-
acteristic power-plant curve.

2. To obtain from the motor its full power and from the
propeller its maximum efficiency the propeller must be
well adapted to the motor, and this altogether independently
of the aeroplane on which they are mounted.

3. There is only a single series of propellers well adapted
to a given motor.

4. For a power-plant to develop maximum efficiency the
aeroplane must fly at a certain speed, known as the best
speed suited to the power-plant under consideration.

FLIGHT IN STILL AIR

TABLE VI.

Pitch
Ratio.

Propeller
Diameters
(in metres).

Propeller
Pitch (in
metres ;
product
of cols. 1
and 2).

Best Suitable Speed.

Maximum
Power-
plant
Efficiency
(from
Table V.)

Maximum
Useful
Power
developed
(product of
50 h.p.
and col. 6).

m.p.s.

km.p.h.

3 4

0-5

2-46

1-23

14-2

0-45

22-5

0-6

2-33

1-40

17-7

0-53

26-5

0-7

2-24

1-57

21-1

0-61

30-5

0-8

2-16

1-73

23-8

0-70

0-9

2'09

1-88

26-4

0-76

1-0

2-04

29-5

106

0-80

1-15

1-98

2-28 33-3

126

0-84

In conclusion, it will be advisable to remember that
the conclusions reached above should not be deemed to
apply with rigorous accuracy. Fortunately, practice is
more elastic than theory. Thus we have already seen in
the case of the angle of incidence of a plane that there is,
round about the value of the best incidence, a certain
margin within whose limits the incidence remains good.
Just so we have to admit that a given power-plant may
yield good results not only when the aeroplane is flying at
a single best speed, but also when its speed does not vary
too widely from this value.

In other words, a certain elasticity is acquired in applying
in practice purely theoretical deductions, though it should
not be forgotten that the latter indicate highly valuable
principles which can only be ignored or thrust aside with
the most serious results, as experience has proved only
too well.

CHAPTER V

FLIGHT IN STILL AIR
THE POWER-PLANT (concluded)

IN the last chapter we confined ourselves mainly to the
working of the power-plant itself, and more particularly to
the mutual relations between its parts, the motor and the
propeller, without reference to the machine they are
employed to propel. The present chapter, on the other
hand, will be devoted to the adaptation of the power-plant
to the aeroplane, and incidentally will lead to some con-
sideration of the variable- speed aeroplane and of the greatest
possible speed variation.

In Chapter II. particular stress was laid on the graph
termed the essential curve of the aeroplane, which enables
us to find the different values of the useful power required
to sustain in flight a given aeroplane at different speeds,
that is, at different angles of incidence and lift coefficients.

In fig. 17 the thin curve (reproduced from fig. 6,
Chapter II.) is the essential aeroplane curve of a Breguet
biplane weighing 600 kg., with an area of 30 sq. m. and
a detrimental surface of 1-2 sq. m.

But in the last chapter particular attention was also
drawn to the graph termed the power-pla-nt curve, which
gives the values of the useful power developed by a given
power-plant when the aeroplane it drives flies at different

In fig. 17 the thick curve is the power-plant curve, in the
case of a motor of 50 h.p. turning at 1200 revolutions per

FLIGHT IN STILL AIR

minute and a propeller of the Chalais-Meudon type, direct-
driven, well adapted to the motor, and with a pitch ratio
of 0-7.

Table VI. (p. 69) gives the diameter and pitch of the
propeller as 2-24 m. and 1-57 m. respectively. The
maximum power-plant efficiency corresponds to a speed of

FIG. 17.

22-1 m. per second. The maximum useful power is 30-5 h.p.
These are the factors which enable us to fix M, the highest
point of the curve.

It will be clear that, by superposing in one diagram (as in
fig. 17, which relates to the specific case stated above) the
two curves representing in both cases a correlation between
useful powers and speeds, and referring, in one case to the
aeroplane, in the other to its power-plant, we should obtain

72 FLIGHT WITHOUT FORMULA

some highly interesting information concerning the adapta-
tion of the power-plant to the aeroplane.

The curves intersect in two points, Rj and R2, which
means that there are two flight speeds, Oj. and O2, at
which the useful power developed by the power-plant is
exactly that required for the horizontal flight of the
aeroplane. These two speeds both, therefore, fulfil the
definition (see Chapter II.) of the normal flying speeds.

From this we deduce that a power-plant capable of
sustaining an aeroplane in level flight can do so at two
different normal flying speeds. But in practice the machine
flies at the higher of these two speeds, for reasons which
will be explained later.

These two normal flying speeds will, however, crop up
again whenever the relation between the motive power and
the speed of the aeroplane comes to be considered. Thus,
when the motive power is zero, that is, when the aeroplane
glides with its engine stopped, the machine can, as already
explained, follow the same gliding path at two different
speeds. The same, of course, applies to horizontal flight,
since, as has been seen, this is really nothing else than an
ordinary glide in which the angle of the flight-path has
been raised by mechanical means, through utilising the
power of the engine.

Let us assume that the ordinary horizontal flight of the
aeroplane is indicated by the point Rl5 which constitutes its
normal flight.

The speed ORj will be roughly 23 m. per second, and the
useful power required, actually developed by the propeller,
about 30 h.p.

According to Table II. (Chapter II.), the normal angle of
incidence will be about 4°, corresponding to a lift coefficient
of 0-038.

Let it be agreed that in flight, which is strictly normal,
the pilot suddenly actuates his elevator so as to increase
the angle of incidence to 6|° (lift coefficient 0-05), and hence
necessarily alters the speed to 20 m. per second.

FLIGHT IN STILL AIR 73

From the thin curve in fig. 17 (and from Table II., on
which it is based) it is clear that the useful power required
to sustain the aeroplane at this speed will be 24 h.p.

On the other hand, according to the thick curve in the
same figure, the power-plant at this same speed of 20 m.
per second will develop a useful power of 30-3 h.p., giving
a surplus of 6-3 h.p. over and above that necessary to
sustain the machine. The latter will therefore climb, and
climb at a vertical speed such that the raising of its weight
absorbs exactly the surplus, NN' or 6-3 h.p., useful power

developed by the power-plant, that is, at a speed of

bOO

=about 0-79 m. per second.

Since this vertical speed must necessarily correspond to a
horizontal speed of 20 m. per second, the angle of the climb,
as a decimal fraction, will be the ratio of the two speeds, i.e.

0-79

=:0<0395=about 4 centimetres per metre=l in 25.

As a matter of fact, we have already seen that by using
the elevator the pilot could make his machine climb or
descend ; but by considering the curves of the aeroplane
and of the power-plant at one and the same time, we gain
a still clearer idea of the process.

Should the pilot increase the incidence to more than 6|°
the speed would diminish still more, and fig. 17 shows that,
in so doing, the surplus power, measured by the distance
dividing the two curves along the perpendicular correspond-
ing to the speed in question, would increase. And with it
we note an increase both in the climbing speed and in the
upward flight-path.

Yet is this increase limited, and the curves show that
there is one definite speed, 01, at which the surplus of
useful power exerted by the power-plant over and above
that required for horizontal flight has a maximum
value.

If, by still further increasing the angle of incidence,
the speed were brought below the limit 01, the climb-

74 FLIGHT WITHOUT FORMULA

ing speed of the aeroplane would diminish instead of
increasing.

Nevertheless, the upward climbing angle would still
increase, but ever more feebly, until the speed attained
another limit, Op, such that the ratio between the climbing
speed to the flying speed, which measures the angle of the
flight-path, attained a maximum.

Thus, there is a certain angle of incidence at which an
aeroplane climbs as steeply as it is possible for it to climb.

If, when the machine was following this flight-path,
the angle of incidence were still further increased by
the use of the elevator, in order to climb still more, the
angle of the flight-path would diminish. Relatively to
its flight-path the aeroplane would actually come down,
notwithstanding the fact that the elevator were set for
climbing.

The same inversion of the effect usually produced from
the use of the elevator would arise if the aeroplane were
flying under the normal conditions represented by the point
R2 in fig. 17. For a decrease in the angle of incidence
through the use of the elevator would have the immediate
and inevitable result of increasing the speed of flight, which
would pass from O2 to Og, for instance. But this would
produce an increase QQ' in the useful power developed by
the power-plant over and above that required for horizontal
flight, so that even though the elevator were set for descend-
ing, the aeroplane would actually climb.

This inversion of the normal effect produced by the
elevator has sometimes caused this second condition of
flight to be termed unstable.

For if a pilot flying hi these conditions, and not aware
of this peculiar effect, felt his machine ascending through
some cause or other, he would work his elevator so as to
come down. But the aeroplane would continue to ascend,
gathering speed the while. The pilot, finding that his
machine was still climbing, would set his elevator still
further for descending until the speed exceeded the limit

FLIGHT IN STILL AIR 75

Op, and the elevator effect returned to its usual state and
the machine actually started to descend. The pilot, unaware
of the existence of this condition and brought to fly under
it by certain circumstances (which, be it added, are purely
hypothetical), would therefore regain normal flight by using
his controls in the ordinary manner.

Nevertheless, one is scarcely justified in applying to this
second condition of horizontal flight the term " unstable "
— if employed in the sense ordinarily accepted in mechanics,
— for one may well believe that a pilot, aware of its
existence, could perfectly well accomplish flight under
this condition by reversing the usual operation of his
elevator.

Still, it would be a difficult proposition for machines
normally flying at a low speed, since the speed of flight
under the second condition (indicated by the point R2, fig.
17) would be lower still.

But in the case of fast machines the solution is obvious
enough. For instance, according to Table II., the minimum
speed of the aeroplane represented by the thin curve in
fig. 17 is about 63 km. per hour, whereas in the early
days of aviation the normal flying speed of aeroplanes
was less.

Now, note that by making an aeroplane fly under the
second condition the angle of the planes would be quite
considerable. In the case in question the angle would be
in the neighbourhood of 15°, which is about 10° in excess of
the normal flying angle.

The whole aeroplane would therefore be inclined at an
angle equivalent to some ten degrees to the horizontal, with
the result that the detrimental surface (which cannot be
supposed constant for such large angles) would be increased,
and with it the useful power required for flight.

In practice, therefore, the power-plant would not enable
the minimum speed Or2 to be attained, and the second condi-
tion of flight would take place at a higher speed and at a
smaller angle of incidence. Still, it would be practicable

76 FLIGHT WITHOUT FORMULA

by working the elevator in the reverse sense to the
usual.*

Now let us just see how a pilot could make his aeroplane
pass from normal flight to the second condition ; although,
no doubt, in so doing we anticipate, for it is highly im-
probable that any pilot hitherto has made such an attempt.

When the aeroplane is flying horizontally and normally,
the pilot would simply have to set his elevator to climb, and
continue this manoeuvre until the flight-path had attained
its greatest possible angle. The aeroplane would then return
(and very quickly too, if practice is in accordance with
theory) to horizontal flight, and now, flying very slowly, it
would have attained to the second condition of flight. At
this stage it would be flying at a large angle to the flight-
path, very cabre, almost like a kite.

The greater part of the useful power would be absorbed
in overcoming the large resistance opposed to forward
motion by the planes. It will now be readily seen that,
under these conditions, any decrease in the angle of incidence
would cause the machine to climb, since, while it would
have but little effect on the lift of the planes, it would
greatly reduce their drag.

By the process outlined above, the aeroplane would
successively assume every one of the series of speeds
between the two speeds corresponding to normal and the
second condition of flight (i.e. it would gradually pass from
Or1 to O2, fig. 17), though it would have to begin with
climbing and descend afterwards.

But we know that the pilot has a means of attaining
these intermediary speeds while continuing to fly horizontally,
namely, by throttling down his engine. This, at all events,
is what he should do until the speed of the machine had

* At present we are only dealing with the sustentation of the aeroplane.
From the point of view of stability, which will be dealt with in subsequent
chapters, it seems highly probable that the necessity of being able to fly at
a small and at a large angle of incidence will lead to the employment of
special constructional devices.

FLIGHT IN STILL AIR

reached a certain point 01 (fig. 18) corresponding to that
degree of throttling at which the power-plant curve (much
flatter now by reason of the throttling-down process) only
continues to touch the aeroplane curve at a single point L.
Below this speed, if the pilot continues to increase the angle
of incidence by using the elevator, horizontal flight cannot
be maintained except by quickly opening the throttle.

It would therefore seem feasible to pass from the normal
to the second condition of flight, without rising or falling,

FIG. 18.

by the combined use of elevator and throttle. But up till
now all this remains pure theory, for hitherto few pilots
know how to vary their speed to any considerable extent,
and probably not a single one has yet reduced this speed
below the point 01 and ventured into the region of the
second condition of flight, that wherein the elevator has to
be operated in the inverse sense.

The reason for this view is that the aeroplane, when
its speed approaches the point 01, is flying without any
margin, and consequently is then bound to descend. If
therefore it obeys the impulse of descending given by the
elevator, it no longer responds to the climbing manipulation.

78 FLIGHT WITHOUT FORMULA

As soon as the pilot perceives this,* he hastens to increase
the speed of his machine again by reducing the angle of
incidence and opening his throttle, whereas, in order to pass
the critical point, he would in fact have to open the throttle
but still continue to set his elevator to climb.

The possibility of achieving several different speeds by
the combined use of elevator and throttle forms the solution
to the problem of wide speed variation.

The greatest possible speed variation which any aeroplane
is capable of attaining is measured by the difference between
the normal and the second condition of flight. But, up to
the present at any rate, the latter has not been reached, and
the lowest speed of an aeroplane is that (indicated by 01,
fig. 18) corresponding to flight at the " limit of capacity."

This particular speed, not to be mistaken for one of the
two essential conditions of flight, is usually very close to
that corresponding to the economical angle of incidence
(see Chapter II.). Hence the economical speed constitutes the
lower limit of variation, which has probably never yet been
attained.

In the future, if the second condition of flight is achieved
in practice, one will be able to fly at the lowest possible
speed an aeroplane can attain. This conclusion may prove
of considerable interest in the case of fast machines, for
any reduction of speed, however slight, is then important.

The highest speed is that of the normal flight of an aero-
plane. In the example represented in fig. 17 this speed is
23 m. per second, or about 83 km. per hour. Since the
economical speed of the machine in question is about 66
km. per hour, the absolute speed variation would be 17 km.
per hour, or, relatively, about 20 per cent. This, however,
is a maximum, since the economical speed, as we know, is
never attained in practice.

The above leads to the conclusion that the way to obtain

* He is the more prone to do this owing to the fact that, with present
methods of design and construction, stability decreases as the angle of
incidence is increased.

FLIGHT IN STILL AIR

a large speed variation is to increase the normal flying
speed.

In the previous example we assumed that the 50 h.p.
motor turning at 1200 revolutions per minute was equipped
with a propeller with a 0-7 pitch ratio, well adapted,
whose characteristic qualities are given in Table VI.

Now let us replace this propeller by another, equally well

rStc]

adapted, but with a pitch ratio of 1-15. According to
Table VI. the diameter of this propeller would be 1-98 m.
and its pitch 2-28 m. The best speed corresponding to
the new propeller would be 33 m. per second, and the
maximum useful power developed at this speed 42 h.p.

Now let the new power-plant curve (thick line) be super-
posed on the previous aeroplane curve (see fig. 19). For
the sake of comparison the previous power-plant curve is
also reproduced in this diagram.

80 FLIGHT WITHOUT FORMULA

The advantage of the step is clear at a glance. In fact,
the normal flying speed increases from Orl — equivalent
to 23 m. per second or 83 km. per hour — to Or\ — equivalent
to 26 m. per second, or about 93 km. per hour. This in-
creases the speed variation from 17 to 27 km. per hour, or
from 20 to 29 per cent.

Again, the maximum surplus power developed by the
power-plant over and above that required merely for
sustentation, amounting to about 7 h.p. with the former
propeller, now becomes about 12 h.p. The quickest climb-

7 x 75
ing speed therefore grows from - — =0-88 m. per second

uOO

12x75

to = 1'5 m. per second.

oOO

Hence, by simply changing the propeller, one obtains the
double result of increasing the normal flying speed of the
aeroplane together with its climbing powers. Nor is the
fact surprising, but merely emphasises our contention that
since highly efficient propellers can be constructed, it will
be just as well to use them.

In order to gain an idea of the relative importance of
increasing the pitch ratio when this ratio has already a
certain value, we may superpose in a single diagram
(fig. 20), on the aeroplane curve, all the power-plant curves
representing the various propellers, well adapted, used with
the same 50-h.p. motor turning at 1200 revolutions per
minute, according to Table VI.

Firstly, it will be evident that a pitch ratio of 0-5 would
not enable the aeroplane in question to maintain horizontal
flight, since the two curves — that of the power-plant and of
the aeroplane — do not meet. In fact, the pitch ratio must
be between 0-5 and 0-6 — 0-54, to be exact — for the power-
plant curve to touch the aeroplane curve at a single point.
Horizontal flight would then be possible, but only at one
speed and without a margin.

But as soon as the pitch ratio increases, the normal flying
speed and the climbing speed increase very rapidly. On

FLIGHT IN STILL AIR

the other hand, once the pitch ratio amounts to 0-9, the
advantage of increasing it still further, though this still
exists, becomes negligible. Beyond 1-0 a further increase
of pitch ratio (in the specific case in question) need not be
considered. All of which are, of course, theoretical con-
siderations, although they point to certain definite principles
which cannot be ignored in practice — a fact of which

;

£+- .

— 0-8

\jti

— -1-00

— — .

7^-0-6
**0-S4^

"0-7 ^
Y>

yjr

^0-J

Speed

offtiyl

tfanffl.j.

erSecJ

0 /

]

J 2
?IG. 20.

0 2

5 J

0 J

constructors, as already remarked, are now becoming
cognisant.

At the same time, the reduction of the diameter necessi-
tated by the use of propellers of great efficiency is not
without its disadvantages, more especially in the case
of monoplanes and tractor biplanes in which the propeller
is situated in front. In these conditions, the propeller
throws back on to the fuselage a column of air which be-
comes the more considerable as the propeller diameter is

FLIGHT WITHOUT FORMULA

reduced, since practically only the portions of the blades
near the tips produce effective work.

It is on this ground that we may account for the fact
that reduction in propeller diameter has not yet, up to a
point, given the good results which theory led one to
expect.

But when the propeller is placed in rear of the machine

Sj teed oj flight

•Sec)

35
FIG. 21.

The figures at the side of the curve indicate the lift.

and the backward flowing air encounters no obstacle, there
is every advantage in selecting a high pitch ratio, and we
have already seen that M. Tatin, in consequence, on his
Torpille fitted a propeller with a pitch exceeding the
diameter.*

* It may also be noticed that the need for reducing the diameter
gradually disappears as the power of the motor increases, because the
diameter of propellers well adapted to a motor increases with the power
of the latter.

FLIGHT IN STILL AIR

The use of propellers of high efficiency, therefore, obviously
increases the speed variation obtainable with any particular
aeroplane.

The lower limit of this speed variation has already been
seen to be the economical speed of the aeroplane.

Now, it should be noted that, in designing high-speed
machines, the use of planes of small camber and with a
very heavy loading has the result of increasing the value
of the economical speed. Thus, the Torpille, already
referred to, appeared to be capable of attaining a speed of
160 km. per hour ; * but its economical speed would have
been about 28 m. per second or 100 km. per hour. Fig. 21
shows, merely for the sake of comparison, the curve of an
aeroplane of this type (weight, 450 kg. ; area, 12-50 sq. m. ;
detrimental surface, 0-30 sq. m.) plotted from the following
table.

TABLE VII.

Speed Value.

fei

1 § £;?

C? c3 ^

11!

gi 13 .3^

H *

** 't? .

1*1

" 5fi gn^,
f° 8 X o

^•s

ill

m.p.s.

km.p.h.

"ill

ill

i*5

Q £

3-x

p<2~

o-oio

216

0-0007

31kg.

87kg.

118 kg.

94 h.p.

0-020

42-4

158

0-0013

0-030

34-6

125

0-0020

0-040

108

0-0034

0-050

26-8

0-0055

0-058

24-9

0-0100

* If we allow it a detrimental surface of 0'30 sq. metre, which is
certainly not enough.

FLIGHT WITHOUT FORMULA

The speed variation of such a machine would be 60 km.
per hour =38 per cent.

If it could fly in the second condition of flight, i.e. at
90 km. per hour, the speed variation would be 70 km. per
hour, or 44 per cent.

In a machine o£ similar type, able to attain a speed of
200 km. per hour (weight, 500 kg. ; area, 9 sq. m. ; detri-

•^

CO
"5

«o ,

«•
o ./

V6_

Speed c

fflykl

{in Tn/x.rS

FIG. 22.

The figures by the side of the curve indicate the lift.

mental surface, 0-03 sq. m.), whose characteristic curve is
plotted in fig. 22, according to Table VIII., the economical
speed would be 34 m. per second, or 125 km. per hour,
giving a speed variation of 75 km. per hour, or 38 per cent.
If it could attain the second condition of flight, i.e. 110 km.
per hour, the variation would be 90 km. per hour, or 45
per cent.

Fortunately, as may be seen, the high-speed machine of

FLIGHT IN STILL AIR

the future should possess a high degree of speed variation.
And in the case of really high speeds even the smallest
advantage in this respect becomes of great importance. It
may well be that the necessity for achieving the greatest

TABLE VIII.

Speed Value.

01 X '*-'

C ~C5 di ^.

iil

|||

s.-s g

PHjg 0<I<

'So ^

fe3^

j/J

g,ix|

ill

!»

III

m.p.s.

km.p.h.

$J!

«l^

p c2

PS*-

o-oio

74-8

270

0-0007

35kg.

134 kg.

169 kg.

168 h.p.

0-020

190

0-0013

101

0-030

43-1

155

0-0020

0-040

37-4

135

0-0034

0-050

33-4

120

0-0055

0-058

111

o-oioo

110

possible speed variation will induce pilots of the extra high
speed machines of the future to attempt, for alighting, to
fly at the second condition of flight.* In this they will
only imitate a bird, which, when about to alight, places its
wings at a coarse angle and tilts up its body.

Fig. 20 further shows that when the pitch ratio is less
than 0-8 the highest point of the power-plant curve lies to
the left of the aeroplane curve. It only lies to the right of
it when the pitch ratio is equal to or greater than 0-9. If
the pitch ratio were 0-85, the highest point of the power-
plant curve would just touch the aeroplane curve, and would
hence correspond to normal flight.

* Attention is, however, drawn to the remarks at the bottom of p. 74.

86 . FLIGHT WITHOUT FORMULA

In Chapter IV. it was shown that the highest point of
the power-plant curve corresponds — the propeller being
supposedly well adapted to the motor — to a rotational
velocity of 1200 revolutions per minute, the normal number
of revolutions at which it develops full power. If, there-
fore, this highest point lies to the left of the aeroplane
curve, the motor is turning at over 1200 revolutions per
minute when the aeroplane is flying at normal speed. On
the other hand, if the highest point lies to the right of the
aeroplane curve, in normal flight the motor will be running
at under 1200 revolutions per minute.

In neither case will it develop full power. Moreover,
there is danger in running the motor at too high a number
of revolutions, particularly if it is of the rotary type. Only
a propeller with a pitch ratio of 0-85 could enable the
motor to develop its full power (in the special case in
question).

This immediately suggests the expedient of keeping the
motor running at 1200 revolutions per minute while allow-
ing the propeller to turn at the speed productive of its
maximum efficiency through some system of gearing.
Thus we are brought by a logical chain of reasoning to the
geared-down propeller, a solution adopted in very happy
fashion in the first successful aeroplane — that of the brothers
Wright.

Let us suppose that an aeroplane whose curve is shown
by the thin line in fig. 23 has a power-plant curve repre-
sented by the thick line in the same figure, the propeller
direct-driven, having a pitch ratio of 1-15, and hence possess-
ing (according to Commandant Dorand's experiments) 84
per cent, maximum efficiency.

Evidently, however good this power-plant might be
when considered by itself, it would be very badly adapted
to the aeroplane in question, since, firstly, it would only
enable the machine to obtain the low speed Orx ; and,
secondly, the maximum surplus of useful power, the measure
of an aeroplane's climbing properties, would fall to a very

FLIGHT IN STILL AIR

low figure. Hence, the machine would only leave the
ground with difficulty, and would fly without any margin.
And all this simply and solely because the best speed, Om,
suited to the power-plant would be too high for the aero-
plane.

Now let the direct-driven propeller be replaced by another
of the same type, but of larger diameter, and geared down
in such fashion that the best speed suited to this power-
plant corresponds to the normal flying speed O'15 of the
aeroplane (see fig. 23).

FIG. 23.

The maximum useful power developed by this power-
plant remains in theory the same as before, since the pro-
peller, being of the same type, will still have a maximum
efficiency of 84 per cent. The new power-plant curve will
therefore be of the order shown by the dotted line in the
figure.

It is clear that by gearing down we first of all obtain an
increase of the normal flying speed, and secondly, a very
large increase in the maximum surplus of useful power —
that is, in the machine's climbing capacity. In practice,
however, this is not a perfectly correct representation of

88 FLIGHT WITHOUT FORMULA

the case, since gearing down results in a direct loss of
efficiency and an increase in weight. Whether or not to
adopt gearing, therefore, remains a question to be decided
on the particular merits of each case. Speaking very
generally, it can be said that this device, which always
introduces some complication, should be mainly adopted in
relatively slow machines designed to carry a heavy load.

In the case of high-speed machines it seems better to
drive the propeller direct, though even here it may yet
prove desirable to introduce gearing.

This study of the power-plant may now be rounded off
with a few remarks on static propeller tests, or bench tests.
These consist hi measuring, with suitable apparatus, on the
one hand, the thrust exerted by the propeller turning at a
certain speed without forward motion, and, on the other,
the power which has to be expended to obtain this result.

Experiment has shown that a propeller of given diameter,
driven by a given expenditure of power, exerts the greatest
static thrust if its pitch ratio is in the neighbourhood of
0-65.* On the other hand, we have seen that the highest
thrust efficiency hi flight is obtained with propellers of a
pitch ratio slightly greater than unity. Hence one should
not conclude that a propeller would give a greater thrust
hi flight simply from the fact that it does so on the bench.
Thus, the propeller mounted on the Tatin Torpille, already
referred to, which gave an excellent thrust hi flight, would
probably have given a smaller thrust on the bench than a
propeller with a smaller pitch.

Consequently, a bench test is by no means a reliable
indication of the thrust produced by a propeller hi flight.
Besides, it is usually made not only with the propeller
alone but with the complete power-plant, in which case the
result is even more unreliable owing to the fact that the
power developed by an internal combustion engine varies
with its speed of rotation.

For instance, suppose that a motor normally turning at
* From Commandant Dorand's experiments.

FLIGHT IN STILL AIR 89

1200 revolutions per minute is fitted with a propeller of
1-15 pitch ratio which, when tested on the bench by itself,
already develops a smaller thrust than a propeller of 0-65
pitch ratio ; the motor would then only turn at 900 revolu-
tions per minute, whereas the propeller of 0-65 pitch ratio
would let it turn at 1000 revolutions per minute, and hence
give more power. The propeller with a high pitch ratio
would therefore appear doubly inferior to the other, and
this notwithstanding the fact that its thrust in flight would
undoubtedly be greater.

A propeller exerting the highest thrust in a bench test must
not for that reason be regarded as the best.

CHAPTER VI
STABILITY IN STILL AIR

LONGITUDINAL STABILITY

AT the very outset of the first chapter it was laid down
that the entire problem of aeroplane flight is not solved
merely by obtaining from the " relative " air current which
meets the wings, owing to their forward speed, sufficient
lift to sustain the weight of the machine ; an aeroplane, in
addition, must always encounter the relative air current in
the same attitude, and must neither upset nor be thrown
out of its path by a slight aerial disturbance. In other
words, it is essential for an aeroplane to remain in equi-
librium ; more, in stable equilibrium.*

We may now proceed to study the equilibrium of an
aeroplane in still air and the stability of this equilibrium.

Since a knowledge of some of the main elementary
principles of mechanics is essential to a proper understand-
ing of the problems to be dealt with, these may be briefly
outlined here.

* The very fact that an aeroplane remains in flight presupposes, as
we have seen, a first order of equilibrium, which has been termed the
equilibrium of sustentation, which jointly results from the weight of the
machine, the reaction of the air, and the propeller-thrust. The mainten-
ance of this state of equilibrium, which is the first duty of the pilot,
causes an aeroplane to move forward on a uniform and direct course.

We are now dealing with a second order of equilibrium, that of the
aeroplane on its flight-path. Both orders of equilibrium are, of course,
closely interconnected, for if in flight the machine went on turning and
rolling about in every way, its direction of flight could clearly not be
maintained uniformly.

STABILITY IN STILL AIR

The most important of these is that relating to the
centre of gravity.

If any body, such as an aeroplane, for instance (fig. 24),
is suspended at any one point, and a perpendicular is
drawn from the point of suspension, it will always pass,
whatever the position of the body in question, through the
same point G, termed the centre of gravity of the body.

The effect of gravity on any body, in other words, the

FIG. 24.

force termed the weight of the body, therefore always
passes through its centre of gravity, whatever position the
body may assume.

Another principle is also of the greatest importance in
considering stability ; namely, the turning action of
forces.

When a force of magnitude F (fig. 25), exerted in the
direction XX, tends to make a body turn about a fixed
point G, its action is the stronger the greater the distance,
Gx, between the point G and the line XX. In other words,
the turning action of a force relatively to a point is the
greater the farther away the force is from the point.

Further, it will be readily understood that a force F',
double the force F in magnitude but acting along a line
YY separated from the fixed point G by a distance Gy,

92 FLIGHT WITHOUT FORMULA

which is just half of Gx, would have a turning force equal
to F. In short, it is the well-known principle of the
lever.

The product of the magnitude of a force by the length
of its lever arm from a point or axis therefore measures
the turning action of the force. In mechanics this turning
action is usually known as the moment or the couple.

When, as in fig. 25, two turning forces are exerted in
inverse direction about a single point or axis, and their

FIG. 25.

turning moment or couple is equal, the forces are said to be
in equilibrium about the point or axis in question.

For a number of forces to be in equilibrium about a point
or axis, the sum of the moments or couples of those acting
in one direction must be equal to the sum of the couples of
those acting in the opposite direction.

It should be noted that in measuring the moment of a
force, only its magnitude, its direction, and its lever arm
are taken into account. The position of the point of its
application is a matter of indifference. And with reason,
for the point of application of a force cannot in any way
influence the effect of the force ; if, for instance, an object

STABILITY IN STILL AIR 93

is pushed with a stick, it is immaterial which end of the
stick is held in the hand, providing only that the force is
exerted in the direction of the stick.

Before venturing upon the problem of aeroplane stability
a fundamental principle, derived from the ordinary theory
of mechanics, must be laid down.

FUNDAMENTAL PRINCIPLE. — So far as the equilibrium
of an aeroplane and the stability of its equilibrium are con-
cerned, the aeroplane may be considered as being suspended
from its centre of gravity and as encountering the relative
wind produced by its own velocity.

This principle is of the utmost importance and absolutely
essential ; by ignoring it grave errors are bound to ensue,
such, for instance, as the idea that an aeroplane behaves in
flight as if it were in some fashion suspended from a certain
vaguely-defined point termed the " centre of lift," usually
considered as situated on the wings. An idea of this sort
leads to the supposition that a great stabilising effect is
produced by lowering the centre of gravity, which is thus
likened to a kind of pendulum.

Now, it will be seen hereafter that in certain cases the
lowering of the centre of gravity may, in fact, produce a
stabilising effect, but this for a very different reason.

The " centre of lift " does not exist. Or, if it exists, it is
coincident with the centre of gravity, which is the one and
only centre of the aeroplane.

The three phases of stability, which is understood to
comprise equilibrium, to be considered are :

Longitudinal stability.

Lateral stability.

Directional stability.

First comes longitudinal stability, which will be dealt
with in this chapter and the next.

Every aeroplane has a plane of symmetry which remains
vertical in normal flight. The centre of gravity lies in
this plane. The axis drawn through the centre of gravity
at right angles to the plane of symmetry may be termed

94 FLIGHT WITHOUT FORMULA

the pitching axis and the equilibrium of the aeroplane
about its pitching axis is its longitudinal equilibrium.

Hereafter, and until stated otherwise, it will be assumed
that the direction of the propeller-thrust passes through
the centre of gravity of the machine. Consequently,
neither the propeller-thrust nor the weight of the aeroplane,
which, of course, also passes through the centre of gravity,
can have any effect on longitudinal equilibrium, for, hi
accordance with the fundamental principle set out above,
the moments exerted by these two forces about the pitch-
ing axis are zero.

Hence, in order that an aeroplane may remain in longi-
tudinal equilibrium on its flight-path, that is, so that it
may always meet the air at the same angle of incidence,
all that is required is that the reaction of the air on the
various parts of the aeroplane should be in equilibrium
about its centre of gravity.

Now, in normal flight all the reactions of the air must
be forces situated in the plane of symmetry of the machine.
These forces may be compounded into a single resultant
(see Chapter II.), which, for the existence of longitudinal
equilibrium, must pass through the centre of gravity.

We may therefore state that : when an aeroplane is
flying in equilibrium, the resultant of the reaction of
the air on its various parts passes through the centre of
gravity.

This resultant will be called the total pressure.

Let us take any aeroplane, maintained in a fixed position,
such, for instance, that the chord of its main plane were at
an angle of incidence of 10°, and let us assume that a hori-
zontal air current meets it at a certain speed.

The air current will act upon the various parts of the
aeroplane and the resultant of this action will be a total
pressure of a direction shown by, say, P10 (fig. 26). Without
moving the aeroplane let us now alter the direction of the
air current (blowing from left to right) so that it meets
the planes at an ever-decreasing angle, passing successively

STABILITY IN STILL AIR

from 10° to 8°, 6°, 4°, etc. In each case the total pressure
will take the directions indicated respectively by P8, P6, P4,
etc. Let G be the centre of gravity of the aeroplane.

Only one of the above resultants — P6, for instance — will
pass through the centre of gravity. From this it may be
deduced that equili-
brium is only possible
in flight when the
main plane is at an
angle of incidence of
6°.

Thus, a perfectly
rigid unalterable
aeroplane could only
in practice fly at a
single angle of in-
cidence.

If the centre of
gravity could be
shifted by some means
or other, to the posi-
tion P4, for instance,
the one angle of in-
cidence at which the
machine could fly
would change to 4°.
But this method for
varying the angle of
incidence has not
hitherto been success-
fully applied in
practice.*

The same result, however, is obtained through an auxil-
iary movable plane called the elevator.

It is obvious that by altering the position of one of the

* It will be seen hereafter that, if the method can be applied, it would
have considerable advantages.

FIG. 2fi.

FLIGHT WITHOUT FORMULA

planes of the machine the sheaf of total pressures is altered.
Thus, figs. 27 and 28 represent the total pressures in the
case of one aeroplane after altering the position of the

FIG. 27.

elevator (the dotted outline indicating the main plane).
If G is the centre of gravity, the normal angle of incidence
passes from the original 4° to 2° by actuating the elevator.

STABILITY IN STILL AIR

Therefore, as stated in Chapter I., by means of the
elevator the position of longitudinal equilibrium of an

FIG. 28.

aeroplane, and hence its incidence, can be varied at
will.

The action of the elevator will be further considered in
the next chapter.

But the longitudinal equilibrium of an aeroplane must

FLIGHT WITHOUT FORMULA

also be stable ; in other words, if it should accidentally lose

its position of equilibrium, the action of the forces arising

through the air current from the very fact of the change

in its position should cause
it to regain this position
instead of the reverse.

If we examine once
again the sheaf of total
pressures we may be able
to gain an idea of how
this condition of affairs
can be brought about.

Returning again to fig.
26, let us suppose that by
an oscillation about its
pitching axis — the move-
ment being counter-clock-
wise — the angle of the
planes, which is normally
6° since the total pressure
P6 passes through the
centre of gravity, decreases
to 4°, the resultant of
pressure on the aeroplane
in its new position will
have the direction P4 ;
hence this resultant will
have, relatively to the
pitching axis, a moment
acting clockwise, which

will therefore be a righting couple since it opposes the

oscillation which called it into being.

The same thing would come to pass if the oscillation was

in the opposite direction.

In this case, therefore, equilibrium is stable.

On the other hand, if the sheaf of pressures was arranged

as in fig. 29, the pressure P4 would exert an upsetting

FIG. 29.

STABILITY IN STILL AIR

En te.rt.rty Ectgre.

0-1

0-2

0-3

0-4

0-5

couple relatively to the pitching axis, and equilibrium
would be unstable.

The stability or instability of longitudinal equilibrium
therefore depends on the relative positions of the sheaf of
total pressures and of the centre of gravity, and it may be laid
down that when the line
of normal pressure is in-
tersected by those of the
neighbouring total pres-
sures at a point about
the centre of gravity,
equilibrium is stable,
whereas it is unstable in
the reverse case.

Several experimenters,
and among them notably
M. Eiffel, have sought to
determine by means of
tests with scale models
the position of the total
pressures corresponding
to ordinary angles of
incidence. Hitherto M.
Eiffel's researches have
been confined to tests on
model wings and not on
complete machines, .but
the latter are now being
employed. Moreover,
the results do not indi- v on , . ,,,

FIG. 30. — Angles t of the chord and the wind.

cate the actual position

and distribution of the pressure itself, but only the point
at which its effect is applied to the plane, this point being
known as the centre of pressure.

The results of these tests have been plotted in two series
of curves which give the position of the centre of pressure
with a change in the angle of incidence. Figs. 30 and 31

0-6

0-7

0-8

0-9

Irt

-400-300-20°-IO° 0° 10° 20° 30

100

FLIGHT WITHOUT FORMULAE

reproduce, by way of indicating the system, the two series
of curves relating to a Bleriot XI. wing.

It has already been remarked that the point from which
a force is applied is of no importance ; accordingly, a centre

-30V

-is*

FIG. 31.

of pressure is of value only in so far as it enables the direc-
tion of the pressures themselves to be traced.

By comparing the curve shown in fig. 31 with the polar
curves already referred to in previous chapters, one obtains

STABILITY IN STILL AIR

101

a means of reproducing both the position and the magnitude,
relatively to the wing itself, of the pressures it receives at
varying

FIG. 32. — Sheaf of pressures on a flat plane.

Figs. 32, 33, and 34 show the sheaf of these pressures in
the case, respectively, of :
A flat plane.

A slightly cambered plane (e.g. Maurice Farman).
A heavily cambered plane (Bleriot XL).
These diagrams, be it repeated, relate only to the plane
by itself and not to complete machines.

* A description of the method may be found in an article published
by the author in La Technique Aeronautique (January 15, 1912).

102

FLIGHT WITHOUT FORMULAE

Comparison of these three diagrams brings out straight
away a most important difference between the flat and the
two cambered planes. That relating to the flat plane,
in fact, is similar in its arrangement to that shown in

FIG. 33. — Sheaf of pressures on a Maurice Farman plane.

fig. 26, which served to illustrate a longitudinally stable
aeroplane.

The diagrams relating to cambered planes, on the other
hand, are analogous, so far as the usual flying angles are
concerned, to fig. 29, which depicted the case of a longi-
tudinally unstable aeroplane.

Thus we can state that, considered by itself, a flat plane
is longitudinally stable, a cambered plane unstable (the

STABILITY IN STILL AIR

103

A -HHffA-

A [in /

FIG. 34. — Sheaf of pressures on a Bleriot XI. plane.

104 FLIGHT WITHOUT FORMULA

latter statement, however, as will subsequently be seen, is
not always absolutely correct). On the other hand, every
one knows nowadays that flat planes are very inefficient,
producing little lift with great drag.

Hence the necessity for finding means to preserve the
valuable lifting properties of the cambered plane while
counteracting its inherent instability. The bird, inciden-
tally, showed that it is possible to fly with cambered wings.
And it was by adopting this example and improving upon
it that the problem was solved, by providing the aeroplane
with a tail.

An auxiliary plane, of small area but placed at a con-
siderable distance from the centre of gravity of the aero-
plane, and therefore possessing a big lever arm relatively
to the centre of gravity, receives from the air, when in
flight the aeroplane comes to oscillate in either direction,
a pressure tending to restore it to its original attitude.
Since this pressure is exerted at the end of a long lever
arm, the couples, which are always righting couples, are of
considerably greater magnitude than the upsetting couples
arising from the inherent instability of the cambered type
itself.

The adoption of this device has rendered it possible to
utilise the great advantage possessed by cambered planes.
Of course it is true that a machine with perfectly flat planes
would be doubly stable, by virtue both of its main planes
and of its tail, but to propel a machine of this type would
mean an extravagant waste of power.

Provided the tail is properly designed, there is nothing
to fear even with an inherently unstable plane, and the
full lifting properties of the camber are nevertheless
retained.

Subsequently it will be shown that the use of a tail
entirely changes the nature of the sheaf of pressures, which,
in an aeroplane provided with a tail, and even though its
planes are cambered, assumes the stable form corresponding
to a flat plane.

STABILITY IN STILL AIR 105

The aeroplane therefore really resolves itself into a main
plane and a tail.*

Assuming, once and for all, that the propeller-thrust
passes through the centre of gravity, the longitudinal
equilibrium of an aeroplane about the centre of gravity
can be represented diagrammatically by one of the three
figs., 35, 36, and 37.

In fig. 35 the tail CD is normally subjected to no pressure
and cuts the air with its forward edge. In this case, equi-
librium exists if the pressure Q (in practice equal to the
weight of the machine) on the main plane AB passes through
the centre of gravity G.

In fig. 36 the tail CD is a lifting tail, that is, normally
it meets the air at a positive angle and therefore is sub-
jected to a pressure q directed upwards. For equilibrium
to be possible in this case the pressure Q on the main plane
AB must pass in front of the centre of gravity G of the
aeroplane, so that its couple about the point G is equal to
the opposite couple q of the tail.

The pressures Q and q must be inversely proportional to
the length of their lever arms. When compounded they
produce a resultant or total pressure equal to their sum
(and to the weight of the aeroplane), which, as we know,
would pass through the centre of gravity.

Lastly, in fig. 37 the tail CD is struck by the air on its
top surface and receives a downward pressure q. To obtain
equilibrium the pressure Q on the main plane AB must
pass behind the centre of gravity G, the couples exerted
about this point by the pressures Q and q being, as before,
equal and opposite. Once again, the pressures Q and q
must be inversely proportional to the length of their lever
arms. If compounded they would produce a resultant
total pressure equal to their difference (and to the weight

* In the case of a biplane both the planes will be considered as forming
only a single plane, a proceeding which is quite permissible and could,
if necessary, be easily justified.

106 FLIGHT WITHOUT FORMULA

G B CD

FIG. 35.

c D

FIG. 36.

P-Q-W /Q

FIG. 37.

STABILITY IN STILL AIR

107

of the aeroplane), which would again pass through the
centre of gravity.

A fourth arrangement (fig. 38), and the first to be adopted
in practice — since the 1903 Wright and the 1906 Santos-
Dumont machines were of this type — is also possible. It
has lately been made use of again hi machines of the
" Canard " type (e.g. in the Voisin hydro-aeroplane), and
consists in placing the tail, which must of course be a
lifting tail, in front of the main plane. The conditions of
equilibrium are the same as in fig. 36.

In an aeroplane, to whichever type it belongs, the term

FIG. 38.

longitudinal dihedral, or Fee, is usually applied to the angle
formed between the chords of the main and tail planes.

Hitherto the relative positions of the main plane and
the tail have been considered only from the point of view
of equilibrium. We have now to consider the stability of
this equilibrium. For this purpose we must return to the
sheaf of pressures exerted, not on the main plane alone,
but on the whole machine, that is, we have to consider the
sheaf of total pressures.

This is shown in fig. 39,* which relates to a Bleriot XI.

* At the time when this treatise was first published, no experiments
had been made to determine the actual sheaf of pressures as it exists in
practice. The accompanying diagrams were drawn up on the basis of
the composition of forces.

108

FLIGHT WITHOUT FORMULA

FIG. 39. — Sheaf of total pressures on a complete Bleriot XI. monoplane.

wing provided with a tail plane of one-tenth the area of
the main plane, making relatively to the main plane a longi-

STABILITY IN STILL AIR 109

tudinal Vee or dihedral of 6°, and placed at a distance
behind the main plane equal to twice the chord of the
latter.

Let it be assumed that the normal angle of incidence of
the machine is 6°, which would be the case if its centre
of gravity coincided with the pressure P6, at G15 for in-
stance.

An idea of the longitudinal stability of the machine in
these conditions may be guessed from calculating the couple
caused by a small oscillation, such as 2°.

Since the normal incidence is 6°, the length of the
pressure P6 is equivalent to the weight of the machine.
By measuring with a rule the length of P4 and P8, it will
be found to be equal respectively to P6xO-74 and to
P6Xl-23. The values of P4 and P8 therefore are the pro-
ducts of the weight of the aeroplane multiplied by 0-74
and 1-23 respectively.

Further, the lever arms of these pressures will, on measure-
ment, be found to be respectively 0-043 and 0-025 times
the chord of the main plane.

By multiplying and taking the mean of the results ob-
tained, which only differ slightly, it will be found that an
oscillation of 2° produces a couple equal to 0-031 times the
weight of the aeroplane multiplied by its chord.

This couple produced by an oscillation of 2° can obviously
be compared to the couple which would be produced by an
oscillation of 2° imparted to the arm of a pendulum or
balance of a weight equal to that of the aeroplane.

For these two couples to be equal, the pendulum arm
must have a length of 0-88 of the chord, or, if the latter be
2m., for instance, the arm would have to measure 1-76 m.
Hence, the longitudinal stability of the machine under
consideration could be compared to that of an imaginary
pendulum consisting of a weight equal to that of the aero-
plane placed at the end of a 1-76 m. arm. It is evident
that the measure of stability possessed by such a pendulum
is really considerable.

110 FLIGHT WITHOUT FORMULA

Having laid down this method of calculating the longi-
tudinal stability of an aeroplane, fig. 39 may once again be
considered.

To begin with, it is evident that if the centre of gravity
is lowered, though still remaining on the pressure line P6,
the longitudinal stability of the machine will be increased
since, the pressure lines being spaced further apart, then-
lever arms will intersect. Therefore, under certain condi-
tions, the lowering of the centre of gravity may increase
longitudinal stability, though this has nothing whatsoever
to do with a fictitious " centre of lift." Besides, in practice
the centre of gravity can only be lowered to a very small
extent, and the possible advantage derived therefrom is
consequently slight, while, on the other hand, it entails
disadvantages which will be dealt with hi the next
chapter.

Finally, the use of certain plane sections robs the lower-
ing of the centre of gravity of any advantages which it
may otherwise possess, a point which will be referred to in
detail hereafter.

Returning to fig. 39 — the normal angle of incidence being
6°, and the non-lifting tail forming this same angle with the
chord of the main plane, the tail plane will normally be
parallel with the wind (see fig. 35).

If the centre of gravity, instead of being at Gl5 were at
G2, on the pressure line P8, the tail would become a lifting
tail (see fig. 36), having a normal angle of incidence of 2°.
Calculating as before, the length of the arm of the imaginary
equivalent pendulum is found to be only 0-63 of the chord,
or 1-26 m. if the chord measures 2 m.

The aeroplane is therefore less stable than in the previous
example.

On the contrary, if the centre of gravity were situated at
G3, corresponding to a normal incidence of 4°, so that the
tail is struck by the wind on its top surface at an angle of
2° (in other words, is placed at a " negative " angle of 2°,
see fig. 37), the equivalent pendulum would have to have

STABILITY IN STILL AIR 111

an arm 3-50 m. long,* or about twice as long as when the
normal incidence is 6°.

From this one would at first sight be tempted to conclude
that the longitudinal stability of an aeroplane is the greater
the smaller its normal flying angle, or, in other words, the
higher its speed ; but, although this may be true in certain
cases, it is not so in others. Thus, if the alteration in the
angle of incidence were obtained by shifting the centre of
gravity, the conclusion would be true, since the sheaf of
total pressures would remain unaltered.

But if the reduction of the angle is effected either by
diminishing the longitudinal dihedral or, and this is really
the same thing, by actuating the elevator, the conclusion no
longer holds good, for the sheaf of total pressures does
change, and in this case, as the following chapter will show,
so far from increasing longitudinal stability, a reduction
of the angle of incidence may diminish stability even to
vanishing point.

It should further be noted that the arrangement shown
diagrammatically in fig. 37, which consists hi disposing
the tail plane so that it meets the wind with its top surface
in normal flight, is productive of better longitudinal stability
than the use of a lifting tail.f This conclusion will be
found to be borne out by fig. 40, showing the pressures
exerted on the main plane by itself.

By measuring the couples, it is clear that if the centre of
gravity is situated at G1? for instance, the plane is unstable,
as we already knew ; but if the centre of gravity were
placed far enough forward relatively to the pressures, at G2,
for instance, a variation in the angle may set up righting
couples even with a cambered plane. The couple resulting
from a variation of this kind is the difference between the

* Actually, the arm is longer if the oscillation is in the sense of a dive
than in the case of stalling, which is quite in agreement with the con-
clusions which will be set out later.

t It will be seen later that this arrangement also seems to be excellent
from the point of view of the behaviour of a machine in winds.

112 FLIGHT WITHOUT FORMULAE

PlS

PIO

FIG. 40.

STABILITY IN STILL AIR

113

couples of the pressure, before and after the oscillation,
about the centre of gravity.

Cambered planes in themselves may therefore be rendered
stable by advancing the centre of gravity.

This is not difficult to understand ; as a plane is further
removed from the centre of gravity it begins to behave

P. 5

FIG. 41. — Sheaf of total pressures on a Maurice Farman aeroplane.

more and more like the usual tail plane. In these conditions
the stability of an aeroplane becomes very good indeed,
since it is assisted by main and tail planes alike.

This explains why the tail-foremost arrangement (see
fig. 38) can be stable, for in this arrangement the tail,
situated in front, really performs the function of an " un-
stabiliser," which is overcome by the inherent stability of

114 FLIGHT WITHOUT FORMULA

the main plane owing to the fact that the latter is situated
far behind the centre of gravity.

Fig. 40 (which relates to the pressures on the main
plane) further shows that if the centre of gravity is low
enough, at G\, for instance, a Bleriot XI. wing would become
stable from being inherently unstable. This is the reason
for the stabilising influence of a low centre of gravity,
which the examination of the sheaf of total pressures
already revealed.

For the sake of comparison, fig. 41 is reproduced, showing
the sheaf of total pressures belonging to an aeroplane of
the type previously considered, but with a Maurice Farman
plane instead of a Bleriot XI. section.

The pressure lines are almost parallel.

Lowering the centre of gravity in a machine of this type
would produce no appreciable advantage.

It will be seen that the pressure lines draw ever closer
together as the incidence increases, and become almost
coincident near 90°. This shows that if, by some means or
other, flight could be achieved at these high angles — which
could only be done by gliding down on an almost vertical
path, the machine remaining practically horizontal, which
may be termed " parachute " flight, or, more colloquially,
a " pancake " — longitudinal stability would be precarious
in the extreme, and that the machine would soon upset,
probably sliding down on its tail. Parachute flight and
" pancake " descents would therefore appear out of the
question, failing the invention of special devices.

CHAPTER VII
STABILITY IN STILL AIR

LONGITUDINAL STABILITY (cone

IN the last chapter it was shown that the longitudinal
stability of an aeroplane depends on the nature of the sheaf
of total pressures exerted at various angles of incidence on
the whole machine, and that stability could only exist if
any variation of the incidence brought about a righting
couple.

But this is not all, for the righting couple set up by
an oscillation may not be strong enough to prevent the
oscillation from gradually increasing, by a process similar
to that of a pendulum, until it is sufficient to upset the
aeroplane.

The whole question, indeed, is the relation between the
effect of the tail and a mechanical factor, known as the
moment of inertia, which measures in a way the sensitive-
ness of the machine to a turning force or couple.

A few explanations in regard to this point may here be
useful.

A body at rest cannot start to move of its own accord.
A body in motion cannot itself modify its motion.

When a body at rest starts to move, or when the motion
of a body is modified, an extraneous cause or force must
have intervened.

Thus a body moving at a certain speed will continue to
move in a straight line at this same speed unless some force
intervenes to modify the speed or deflect the trajectory.

116

FLIGHT WITHOUT FORMULA

The effect of a force on a body is smaller, the greater the
inertia or the mass of the latter.

Similarly, if a body is turning round a fixed axis, it will
continue to turn at the same speed unless a couple exerted
about this axis comes to modify this speed.

This couple will have the smaller effect on the body, the
more resistance the latter opposes to a turning action, that
is, the more inertia of rotation it possesses. It is this
inertia which is termed the moment of inertia of the body
about its axis. The moment of inertia increases rapidly as
the masses which constitute the body are spaced further
apart, for, in calculating the moment of inertia, the dis-
tances of the masses from the axis of rotation figure, not

in simple proportion, but as their square. An example will
make this principle, which enters into every problem con-
cerning the oscillations of an aeroplane, more clear.

At O, on the axis AB (fig. 42) of a turning handle a rod
XX is placed, along which two equal masses MM can slide,
their respective distances from the point O always remain-
ing equal. Clearly, if the rod, balanced horizontally, were
forced out of this position by a shock, the effect of this
disturbing influence would be the smaller, the further the
masses MM were situated from the point O, in other words,
the greater the moment of inertia of the system.

If the rod were drawn back to a horizontal position by
means of a spring it would begin to oscillate ; these oscilla-
tions will be slower the further apart the masses ; but, on
the other hand, they will die away more slowly, for the

STABILITY IN STILL AIR 117

system would persist longer in its motion the greater its
moment of inertia.

These elementary principles of mechanics show that an
aeroplane with a high moment of inertia about its pitching
axis, that is, whose masses are spread over some distance
longitudinally instead of being concentrated, will be more
reluctant to oscillate, while its oscillations will be slow, thus
giving the pilot time to correct them. On the other hand,
they persist longer and have a tendency to increase if the
tail plane is not sufficiently large.

This relation between the stabilising effect of the tail
and the moment of inertia in the longitudinal sense has
already been referred to at the beginning of this chapter.
It may be termed the condition of oscillatory stability.

In practice most pilots prefer to fly sensitive machines
responding to the slightest touch of the controls. Hence
the majority of constructors aim at reducing the longi-
tudinal moment of inertia by concentrating the masses.

It should be added that the lowering of the centre of
gravity increases the moment of inertia of an aeroplane
and hence tends to set up oscillation, one of the disadvan-
tages of a low centre of gravity which was referred to in
the last chapter.

By concentrating the masses the longitudinal oscillations
of an aeroplane become quicker and, although not so easy
to correct, present one great advantage arising from their
greater rapidity.

For, apart from its double stabilising function, the tail
damps out oscillations, forms as it were a brake in this
respect, and the more effectively the quicker the oscillations.
The reason for this is simple enough. Just as rain, though
falling vertically, leaves an oblique trace on the windows
of a railway-carriage, the trace being more oblique the
quicker the speed of travel, so the relative wind caused by
the speed of the aeroplane strikes the tail plane at a
greater or smaller angle when the tail oscillates than when
it does not, and this with all the greater effect the quicker

118 FLIGHT WITHOUT FORMULAE

the oscillation. It is a question of component speeds
similar to that which will be considered when we come to
deal with the effect of wind on an aeroplane.

The oscillation of the tail therefore sets up additional
resistance, which has to be added to the righting couple due
to the stability of the machine, as if the tail had to move
through a viscous, sticky fluid, and this effect is the more
intense the quicker the oscillation. It is a true brake effect.

In this respect the concentration of the masses possesses
a real practical advantage.

According to the last chapter, an entirely rigid aeroplane,
none of whose parts could be moved, could only fly at a
single angle, that at which the reactions of the air on its
various parts are in equilibrium about the centre of gravity.
In order to enable flight to be made at varying angles the
aeroplane must possess some movable part — a controlling
surface.

Leaving aside for the moment the device of shifting the
centre of gravity (never hitherto employed), the easiest
method would be to vary the angle formed by the main
plane and the tail, i.e. the longitudinal dihedral.

The method was first adopted by the brothers Wright,
and is even at the present time employed in several
machines. Very powerful in its effect, the variations in
the angle of the tail plane affect the angle of incidence by
more than then' own amount, and this hi greater measure
the bigger the angle of incidence.

Figs. 43 and 44 represent two different positions of the
sheaf of total pressures on an aeroplane with a Bleriot XI.
plane, and a non-lifting tail of an area one-tenth that of
the mam plane and situated in rear of it at a distance equal
to twice the chord. In fig. 43 the tail plane forms an
angle of 8° with the chord of the main plane ; in fig. 44
this angle is only 6°.

If the centre of gravity is situated at Gx, the normal
angle of incidence passes from 4° in the first case to 2° in
the second. This variation in the angle of incidence is

STABILITY IN STILL AIR

119

FIG. 43. — Sheaf of total pressures on a Bleriot XI. monoplane with a
longitudinal V of 8°.

120

FLIGHT WITHOUT FORMULA

PlO
P8

ill

FIG. 44. — Sheaf of total pressures on a Bleriot XI. monoplane with a
longitudinal V of 6°.

STABILITY IN STILL AIR 121

therefore integrally the same as that of the angle of the
tail plane.

If the centre of gravity is at G2, the normal angle of
incidence would pass from 6° to 3|°, and would therefore
vary by 2|° for a variation in the angle of the tail of
only 2°.

Lastly, if the centre of gravity is at G3, the normal angle
of incidence would pass from 8° to 5°, a variation equal to
one and a half times that of the angle of the tail.

A comparison of figs. 43 and 44 further shows that the
lines of total pressure are spaced further apart the greater
the longitudinal dihedral. Now, other things being equal,
the farther apart the lines of pressure the greater the longi-
tudinal stability of an aeroplane. Hence the value of the
longitudinal dihedral is most important from the point of
view of stability.

If the tail plane (non-lifting) is normally parallel to the
relative wind, the longitudinal dihedral is equal to the
normal angle of incidence. But if a lifting tail is employed,
the longitudinal dihedral must necessarily be smaller than
the angle of incidence (this is clearly shown in fig. 36).
If the normal angle of incidence is small, as in the case of
large biplanes and high-speed machines, the longitudinal
dihedral is very small indeed and stability may reach a
vanishing point.

But if, in normal flight, the tail plane meets the wind
with its upper surface (i.e. flies at a negative angle), the
longitudinal dihedral, however small the normal angle of
incidence, will always be sufficient to maintain an excellent
degree of stability. This conclusion may be compared
with that put forward in the previous chapter in regard
to the advantage of causing the tail to fly at a negative
angle.

The foregoing shows that the reduction of the angle of
incidence by means of a movable tail plane — i.e. by alter-
ing the longitudinal dihedral — has the disadvantage that
every alteration in the position of the tail plane brings

122 FLIGHT WITHOUT FORMULAE

about a variation in the condition of stability of the
aeroplane.

By plotting the sheaf of total pressures corresponding
to very small values of the longitudinal dihedral, it would
soon be seen that if the latter is too small, equilibrium
may become unstable.

A machine with a movable tail and normally possessing
but little stability — such, for instance, as a machine whose
tail lifts too much — may lose all stability if the angle of
incidence is reduced for the purpose of returning to earth.
This effect is particularly liable to ensue when, at the
moment of starting a glide, the pilot reduces his incidence,
as is the general custom.

Losing longitudinal stability, the machine tends to
pursue a flight-path which, instead of remaining straight,
curls downwards towards the ground, and at the same time
the speed no longer remains uniform and is accelerated.

The glide becomes ever steeper. The machine dives,
and frequently the efforts made by the pilot to right it by
bringing the movable tail back into a stabilising position
are ineffectual by reason of the fact that the tail becomes
subject, at the constantly accelerating speed, to pressures
which render the operation of the control more and more
difficult.

In the author's opinion, the use of a movable tail is
dangerous, since the whole longitudinal equilibrium depends
on the working of a movable control surface which may
be brought into a fatal position by an error of judgment,
or even by too ample a movement on the part of the pilot.

For, apart from the case just dealt with, should the
movable tail happen to take up that position in which
the one angle of incidence making for stability is that
corresponding to zero lift, i.e. when the main plane meets
the wind along its " imaginary chord " (see Chapter I.),
longitudinal equilibrium would disappear and the machine
would dive headlong.

In this respect, therefore, the movements of a movable

STABILITY IN STILL AIR 123

tail should be limited so that it could never be made to
assume the dangerous attitude corresponding to the rupture
or instability of the equilibrium.

A better method is to have the tail plane fixed and rigid,
and, hi order to obtain the variations in the angle of in-
cidence required in practical flight, to make use of an
auxiliary surface known as the elevator.

Take a simple example, that of the aeroplane diagram-
matically shown in fig. 45, possessing a non-lifting tail

C D £

FIG. 45.

plane CD, normally meeting the wind edge-on, to which is
added a small auxiliary plane DE, constituting the elevator,
capable of turning about the axis D.

So long as this elevator remains, like the fixed tail,
parallel to the flight-path, the equilibrium of the aeroplane
will remain undisturbed. But if the elevator is made to
assume the position DE (fig. 46), the relative wind strikes
its upper surface and tends to depress it. Hence the
incidence of the main plane will be increased until the
couple of the pressure Q exerted about the centre of gravity,
and the couple of the pressure q' exerted on the elevator,
together become equal to the opposite moment of the
pressure q on the fixed tail.

Again, if the elevator is made to assume the position

124

FLIGHT WITHOUT FORMULAE

DE2 (fig. 47), the incidence decreases until a fresh condition
of equilibrium is re-established.

Each position of the elevator therefore corresponds to

FIG. 46.

one single angle of incidence ; hence the elevator can be
used to alter the incidence according to the requirements
of the moment.

FIG. 47.

It will be obvious that the effectiveness of an elevator
depends on its dimensions relatively to those of the fixed
tail, and, further, that if small enough it would be incapable,
even in its most active position, to reduce the angle of

STABILITY IN STILL AIR 125

incidence to such an extent as to break the longitudinal
equilibrium of the aeroplane.

This, in the author's opinion, is the only manner in
which the elevator should be employed, for the danger of
increasing the elevator relatively to the fixed tail to the
point even of suppressing the latter altogether has already
been referred to above.

In the position of longitudinal equilibrium corresponding
to normal flight, the elevator, in a well-designed and well-
tuned machine, should be neutral (see fig. 45). It follows
that all the remarks already made with reference to the
important effect on stability of the value of the longitudinal
dihedral apply with equal force when the movable tail
has been replaced by a fixed tail plane and an elevator.

The extent of the longitudinal dihedral depends on the
design of the machine, and more especially on the position
of the centre of gravity relatively to the planes, and on
its normal angle of incidence, which, again, is governed by
various factors, and in chief by the motive power.

The process of tuning-up, just referred to, consists prin-
cipally in adjusting by means of experiment the position of
the fixed tail so that normally the elevator remains neutral.
Tuning-up is effected by the pilot ; in the end it amounts to
a permanent alteration of the longitudinal dihedral ; where-
fore attention must be drawn to the need for caution in
effecting it.

There are certain pilots who prefer to maintain the
longitudinal dihedral rather greater than actually necessary
(i.e. with the arms of the V close together), with the con-
sequence that their machines normally fly with the elevator
slightly placed in the position for coming down, or meeting
the wind with its upper surface. In the case of machines
with tails lifting rather too much, the practice is one to
be recommended, for machines of this description are
dangerous even when possessing a fixed tail, since if the
elevator is moved into the position for descent the longi-
tudinal dihedral is still diminished, though in a lesser

126 FLIGHT WITHOUT FORMULA

degree, and if it were already very small, stability would
disappear and a dive ensue.

Therefore the tuning-up process referred to has this
advantage in the case of an aeroplane with a fixed tail
exerting too much lift, that it reduces the amplitude of
dangerous positions of the elevator and increases the
amplitude of its righting positions.

If the size of the elevator is reduced, with the object of
preventing loss of longitudinal equilibrium or stability,
to such a pitch as to cause fear that it would no longer
suffice to increase the angle of incidence to the degree
required for climbing, an elevator can be designed which
would act much more strongly for increasing the angle
than for reducing it, by making it concave upwards if
situated in the tail, or concave downwards if placed in
front of the machine.

For it may be placed either behind or in front, and
analogous diagrams to those given in figs. 46 and 47 would
show that its effect is precisely the same in either case.

But it should also be noted that if an elevator normally
possessing no angle of incidence is moved so as to produce
a certain variation in the angle of incidence of the main
plane, of 2°, for instance, the angle through which it must
be moved will be smaller in the case of a front elevator
than in that of a rear elevator, the difference between the
two values of the elevator angle being double (i.e. 4° in the
above case) that of the variation in the angle of incidence
(assuming, of course, that front and rear elevators are of
equal area and have the same lever arm).

This is easily accounted for by the fact that a variation
in the angle of incidence, which inclines the whole machine,
is added to the angular displacement of a front elevator,
whereas it must be deducted from that of the rear
elevator.

Thus, if we assume that the elevator must be placed at
an angle of 10° to cause a variation in the incidence of 2°,
the elevator need only be moved through 8° if placed in

STABILITY IN STILL AIR 127

front, whereas it would have to be moved through 12° if
placed in rear.

A front elevator, therefore, is stronger in its action than
a rear elevator. But it is also more violent, as it meets the
wind first, which may tend to exaggerated manoeuvres.
Finally, referring to the remarks in the previous chapters
regarding the " tail-first " arrangement, the longitudinal
stability of an aeroplane is diminished to a certain degree
when the elevator is situated in front. These are no doubt
the reasons that have led constructors to an ever-increasing
extent to give up the front elevator.*

All these facts plainly go to show, as already stated, that
stability does not necessarily increase with speed. Aero-
planes subject to a sudden precipitate diving tendency only
succumb to it when their incidence decreases to a large
extent and their speed exceeds a certain limit, sometimes
known as the critical speed, at which longitudinal stability,
far from increasing, actually disappears altogether. The
term critical speed is not, however, likely to survive long,
if only because it refers to a fault of existing machines
which, let us hope, will disappear in the future. And it
would disappear all the more rapidly if the variations in
the angle of incidence required in practical flight could be
brought about, not by a movable plane turning about a
horizontal axis, but by shifting the position of the centre of
gravity relatively to the planes, which could be done by
displacing heavy masses (such as the engine and passengers'
seats, for example) on board or, also, by shifting the planes
themselves.

In this case, as we have seen, the variations of the in-
cidence would have no effect on the longitudinal dihedral,
so that the sheaf of total pressures would not change, and
then it would be true that stability increased with the
speed. Then, also, there would be no critical speed.

* The placing of the propeller in front and the production of tractor
machines — though, in the author's opinion, an unfortunate arrangement
— has also formed a contributory cause.

128 FLIGHT WITHOUT FORMULA

As stated previously, the horizontal flight of an aeroplane
is a perpetual state of equilibrium maintained by con-
stantly actuating the elevator. The idea of controlling
this automatically is nearly as old as the aeroplane itself.
But, as this question of automatic stability chiefly arises
through the presence of aerial disturbances and gusts, its
discussion will be reserved for the final chapter, which
deals with the effects of wind on an aeroplane.

Hitherto it has been assumed that the propeller-thrust
passes through the centre of gravity, and therefore has
no effect on longitudinal equilibrium. The angle of inci-
dence corresponding to a given position of the elevator
therefore remains the same in horizontal, climbing, or
gliding flight.

But if the propeller-thrust does not pass through the
centre of gravity, it will exert at this point a couple which,
according to its direction, would tend either to increase or
diminish the incidence which the aeroplane would take up
as a glider (assuming that the elevator had not been moved).
In that case any variation in the propeller- thrust, more
particularly if it ceased altogether either by engine failure
or through the pilot switching off, would alter the angle of
incidence.

Thus if the thrust passed below the centre of gravity
the stopping of the engine would cause the angle of in-
cidence to diminish, and thus produce a tendency to dive.
On the other hand, if the thrust is above the centre of
gravity, the stopping of the engine would increase the
angle of incidence, and therefore tend to make the machine
stall.

Practical experience with present-day aeroplanes teaches
that in case of engine stoppage it is better to decrease the
angle of incidence than to leave it unchanged, and, above
all, than to increase it.

The reason for this is that the transition from horizontal
flight to gliding flight is not instantaneous as is often
thought from purely theoretical considerations. An aero-

STABILITY IN STILL AIR 129

plane moving horizontally tends, through its inertia, to
maintain this direction. Since there is now no longer any
propeller-thrust to balance the head-resistance of the
machine, it loses speed, which is to be avoided at all
costs by reason of the ensuing dive. Therefore a pilot
reduces his angle of incidence in order to diminish the
drag of the aeroplane, and hence to maintain speed as far
as possible.

This action usually produces the desired effect, as the
normal angle of incidence of most aeroplanes is greater
than their optimum angle ; but this would not be the case if
the optimum angle, or a still smaller angle, constituted the
normal flying angle.

The reduction of the angle of incidence at the moment
the engine stops has the additional effect of producing the
flattest gliding angle, which, as has already been shown,
corresponds to the use of the optimum angle. On the
other hand, stability increases through the reduction of
the incidence (which is here equivalent to an increase
in speed) so long as this does not reduce the longitudinal
dihedral.

Bearing these various considerations in mind, it would
seem preferable, in contradiction to a very general view
which at one time the author shared, to make the propeller-
thrust pass below rather than above the centre of gravity,
at any rate in the case of machines normally flying at a
fairly large angle of incidence.

As a general rule the propeller-thrust passes approxi-
mately through the centre of gravity, and this, perhaps, is
the best solution of all.

Since the direction of the propeller-thrust is under con-
sideration, it may be as well to note that this direction need
not necessarily be that of the flight-path of the aeroplane.
Take the case where the thrust passes through the centre
of gravity ; it will be readily understood that if the direction
of the thrust is altered this cannot have any effect on
longitudinal equilibrium. Hence there is no theoretical

130 FLIGHT WITHOUT FORMULAE

reason why an aeroplane with an inclined propeller shaft
should not fly horizontally.

The only effect on the flight of an aeroplane by tilting
the propeller shaft up at an angle would be to reduce the
speed, because the thrust doing its share in lifting, the planes
need only exert a correspondingly smaller amount of lift.
Therefore the lifting of the propeller shaft virtually amounts
to diminishing the weight of the aeroplane, thereby, other
things being equal, reducing the speed.

If the thrust became vertical, the planes could be dispensed
with, horizontal speed would disappear, and the aeroplane
would become a helicopter.

It can easily be shown that the most advantageous
direction to give to the propeller-thrust is that wherein the
shaft is slightly inclined upwards, as is done hi the case of
certain machines, though in others the thrust is normally
horizontal.

To wind up these remarks on longitudinal stability, we
will describe various types of little paper gliders which
will afford in practical fashion some interesting information
concerning certain aspects of longitudinal equilibrium and
of gliding flight. The results, of course, are only approxi-
mate in the widest sense, since such paper gliders are very
erratic as they do not preserve their shape for any length
of time.

Experiments with these little paper models are most
instructive and are to be highly recommended to every
reader ; however childish they may at first appear, they
will not be waste of time. By experimenting oneself with
such miniature flying-machines one can learn many valuable
lessons in regard to points of detail, only a few of which
can here be set out. To make these little models it is best
to use the hardest obtainable paper, though it must not be
heavy ; Bristol-board will serve the purpose. Even better
is thin sheet aluminium about one- tenth of a millimetre in
thickness, but in this case the dimensions given hereafter
should be slightly increased.

STABILITY IN STILL AIR 131

TYPE I.

An ordinary rectangular piece of paper, in length about
twice the breadth (12 cm. by 6 cm., for instance), folded
longitudinally down the centre (see fig. 48) so as to form a
very open angle (the function of this, which affects lateral
stability, will be explained in the next chapter).

Reference to fig. 32, Chapter VI., will show that for a
single flat plane to assume one of the ordinary angles of
incidence (roughly, from 2° to 10°), its centre of gravity
must be situated at a distance of from one-third to one-
quarter of the fore-and-aft dimension of the plane from the
forward edge. This is easily obtained by attaching to

FIG. 48. — Perspective.

the paper a few paper clips or fasteners, fixed near one of
the ends of the central fold at a slight distance from the
edge (about | cm.).

If the ballasted paper is held horizontally by its rear end
and is thrown gently forward, it will behave in one of the
three following ways : —

(a) The paper inclines itself gently and glides down
regularly without longitudinal oscillations.

This is the most favourable case, for at the first attempt
the ballast has been placed in the position where the corre-
sponding single angle of incidence was one of the usual
angles. Practice therefore confirms theory, which taught
that a single flat plane is longitudinally stable.

(b) The paper dips forward and dives.

The centre of gravity is too far forward and in front
of the forward limit of the centre of pressure. To obtain

132 FLIGHT WITHOUT FORMULA

a regular glide the ballast must be moved slightly toward
the rear. In effecting this, it will probably be moved too
far back and the paper will in that case behave in the
opposite manner, which is about to be described.

(c) The paper at first inclines itself, but, after a dive
whose proportions vary with several factors, and chiefly
with the force with which the model has been thrown, it
rears up, slows down, and starts another dive bigger than
the first, and thus continues its descent to the ground, stall-
ing and diving in succession (see fig. 49).

FIG. 49.

As a matter of fact, the dive following the first stalling
may be final and become vertical if during the accompany-
ing oscillation the paper should meet the air edge-on, so
that actually it has no angle of incidence, for such a glider
if dropped vertically, leading edge down, has no occasion
to right itself and continues to fall like any solid body.

The above experiment is quite instructive. It corresponds
to the case where the single angle of incidence at which
flight is possible, owing to the centre of gravity being too
far back, is greater than the usual angles of incidence.

As it begins its descent the sheet of paper, having been
thrown forward horizontally, has a small angle of incidence,
and hence tends to acquire the fairly high speed correspond-
ing to this small angle. But the pressure of the air, passing

STABILITY IN STILL AIR 133

in front of the centre of gravity, produces a stalling couple
which increases the incidence. Owing to its inertia, the
paper will tend to maintain its speed, which has now be-
come higher than that corresponding to its large angle
of incidence, and so the pressure of the air becomes greater
than the weight, on account of which the flight-path becomes
horizontal again and even rises.

The same thing, in fact, always happens if for some
reason or other a glider or an aeroplane should attain to
a higher speed than that corresponding to the incidence
given it by the elevator, and also if the angle of the planes
is suddenly increased. This rising flight-path by an in-
crease in the angle of incidence is constantly followed by
birds, and especially by birds of prey such as the falcon,
which uses it to seize its prey from underneath.

Pilots also use it in flattening out after a steep dive or
vol pique, though the manoeuvre is distinctly dangerous,
since it may produce in the machine reactions of inertia
which may cause the failure of certain parts of the
structure.

Returning to the ballasted sheet of paper : as the flight-
path rises, the glider loses speed ; in fact, it may stop
altogether. It is then in the same condition as if it were
released without being thrown forward, and falls in a
steep dive which, as already stated, may prove final.

There are many variants of the three phenomena described.

Thus, the stalling movement may become accentuated to
such an extent as to cause the sheet of paper to turn right
over and " loop the loop." * Again, the paper may start to
glide down backwards and do a " tail-slide."

These variants depend mainly on how far back the
centre of gravity is situated, that is, on the value of the
single angle of incidence at which the sheet can fly. If

* It is interesting to note that this and many of the following
mano3uvres are precisely those practised by Pegoud and his imitators,
although the above was written long before they were attempted in
practice. — TKANSLATQR.

134 FLIGHT WITHOUT FORMULA

this angle is only slightly greater than the usual angles of
incidence, the stability of the glider — which is less, of
course, at large angles than at small ones — will still be
sufficient to prevent the effect of inertia of oscillation from
bringing it into a position where it is liable to dive, to turn
over on its back, or slide backwards. It will therefore
follow a sinuous flight-path consisting of successive stalling
and diving, but will not actually upset.

But if the centre of gravity is brought further back and
the angle of incidence corresponding to this position is
much greater than the usual angles of incidence, the stabilis-
ing couples no longer suffice to overcome the effects of
inertia to turning forces, the condition of stability in
oscillation is no longer fulfilled, and the glider behaves in
one of the ways already described.

It should, however, be pointed out that a rectangular
sheet of paper has a far larger moment of inertia in respect
to pitching than a glider generally conforming, as our next
models will do, to the shape of an aeroplane.

To prevent these occurrences from taking place, all that
is required is to bring the ballast further forward and to
adjust the incidence by cutting off thin strips from the
forward edge. By these means it is possible eventually
to obtain a regular gliding path without longitudinal
oscillations.

If thin strips of paper are thus cut off with sufficient
care,* the various properties of gliding flight set forth in
Chapter II. can be very easily followed.

It will be seen that by gradually reducing the angle
of incidence by cutting back the forward edge, the glide
becomes both longer and faster. Next, when the angle
has become smaller than the optimum angle of this embryo
glider, the length of the glide diminishes, the path becomes
steeper, and the glider tends to dive.

Towards the end the process of adjustment becomes

* In case the ballast should be in the way, the paper can be cut away
diagonally and equally on either side, as shown in fig. 50.

STABILITY IN STILL AIR 135

exceptionally delicate, for since the optimum angle of a
model of this nature is very small indeed, by reason of the
fact that its detrimental surface is almost zero relatively
to its lifting area, the slightest shifting of the centre of
gravity is enough to cause a large variation in the gliding
angle and to upset longitudinal stability.

Now let us suppose that, the ballast being so placed that
the glider tends to dive, we proceed to rectify by cutting
away pieces of the trailing edge as in fig. 50. If the outer
rear tips thus symmetrically formed are bent upwards,

FIG. 50.

the glider will no longer tend to dive and will assume a
position of equilibrium.

By bending these outer tips through various degrees, and
also, if necessary, bending up the inner portion of the
trailing edge, all the various forms of gliding flight can be
reproduced which were previously obtained by shifting the
ballast and cutting back the forward edge.*

But to whatever degree the tips may be bent up, hence-
forward the stalling movement will not be followed by a
dive, nor will the glider loop the loop or do a tail-slide.

This is due to the fact that instead of being constituted
by a'single flat plane, the glider now possesses a tail, which
gives it much better longitudinal stability. The effects of

* The rear tips may not be bent exactly equally on either side, with
the result that the glider may tend to swerve to left or right. To counter-
act this, the tip on the side towards which the paper swerves should be
bent up a little more.

136 FLIGHT WITHOUT FORMULA

inertia are now overcome by the stabilising moments arising
from the tail. Moreover, a glider of this description when
dropped vertically rights itself. It can no longer dive
headlong.

If the tips are bent back to their original horizontal
position, it is evident that the sheet of paper will dive once
more, and to an even greater extent if the tips were bent
down instead of up. This plainly shows the danger of
allowing the elevator to constitute the solitary tail plane,
for, unless its movement is limited, it could cause equili-
brium to be lost.

TYPE II.

1. Fold a sheet of paper in two, and from the folded
paper cut out the shape shown in fig. 51.

2. Fold back the wings and the tail plane along the
dotted lines. The wings should make a slight lateral V or
dihedral.

3. Ballast the model somewhere about the point L — the
exact spot must be found by experiment — with one or
more paper fasteners.

This model approaches more nearly to the usual shape of
an aeroplane. By finding the correct position for the
ballast, so that the centre of gravity is situated on the
total pressure line corresponding approximately to the
optimum angle, this little glider can be made to perform
some very pretty glides.*

The ballast may be brought further forward or additional
paper fasteners may be affixed without making the model
dive headlong.

It will dive, and on this account may be brought to fall
headlong if the height above the ground is only slight ; but
if there is room enough it will recover and, though coming
down steeply, will not fall headlong. It is still gliding,

* Should it tend to swerve to either side, bend up slightly the rear
tip of the wing on the opposite side of that towards which the aeroplane
tends to turn.

STABILITY IN STILL AIR 137

since during its descent the air still exerts a certain amount
of lift. Longitudinal equilibrium is not upset, and if the
glider does not lose its proper shape on account of its high
speed, it cannot fall headlong, whatever the excess of load
carried, by reason of the fact that the main and tail planes
are placed at an angle to one another.

The reduction in the angle of incidence by bringing the
centre of gravity further forward therefore maintains
stability, and even increases it as the speed grows. And
this because the longitudinal dihedral has not been touched.

By shifting the ballast toward the rear, the model will

FIG. 51.

also follow a steep downward path, but this time the angle
of incidence is large, the speed slow, and therefore the glider
remains almost horizontal and " pancakes." This shows
conclusively that the same gliding path can be followed at
two different normal angles of incidence and at two different
speeds.

By still shifting the ballast farther back, the model may
be made to glide as if it belonged to the tail-foremost or
" Canard " type (cf. the third model described hereafter).
Flight at large angles of incidence is now possible and
will not cause the model to overturn as in the case of the
single sheet of paper, as the moment of pitching inertia
is much feebler than in the former case. The stability of

138 FLIGHT WITHOUT FORMULA

oscillation is therefore still adequate at large angles of
incidence.

Now let us shift the ballast back again so that the glide
becomes normal once more ; at the rear of the tail plane,
bend down either the whole or half the trailing edge to
the extent of 2 mm. This will give us an elevator, while
the fixed tail is retained.

By moving this elevator the conditions of gliding flight
can obviously be modified ; for instance, if the outer halves
of the rear edge are evenly bent down to an angle of some
45° — that is, to have their greatest effect in reducing the
angle of incidence — the glider will extend the length of
its flight and travel faster (see fig. 52).

FIG. 52.— Perspective.

But it will still be impossible by the operation of the
elevator to make the model fall headlong. The fixed tail
will prevent this, and will overcome the action of the
elevator because the latter is small in extent. Hence, an
elevator small enough relatively to the tail plane cannot make
an aeroplane dive headlong.

If the whole of the trailing edge is bent down it might
possibly cause longitudinal equilibrium to be upset and
make the glider dive. And should this not prove to be
the case, it could be done without fail by increasing the
depth of the elevator.

The experiment shows that the size of the elevator
should not be too large ; it should merely be sufficient to
cause the alterations of the angle of incidence required for
ordinary flight and should never be able to upset stability.

STABILITY IN STILL AIR 139

TYPE III.

1. Cut out from a sheet of paper folded in two a piece
shaped as in fig. 53.

2. Fold back the wings along the dotted lines.

3. Fold the wing- tips upwards along the outer dotted
line.

This tail-first glider will be stable without ballast and

FIG. 53.

glides very prettily on account of its lightness. It will be
referred to again in connection with directional stability.*

TYPE IV.

1. Cut out from a sheet of paper folded in two the shape
shown in fig. 54.

2. Cut away from the outer edge of the fold two portions
about 1 mm. deep, and of the length shown at AB and CD.

3. Inside the fold fix with glue —

(a) At AB a strip of cardboard or cut from a visiting
card ; 5 cm. long, 1 cm. broad. The inner end
of the strip is shown by the dotted line at AB.

(6) At CD glue a similar strip as shown.

* If it tends to swerve, slightly bend the whole of the front tail in
the opposite sense.

140

FLIGHT WITHOUT FORMULAE

4. Fold back the wings and the tail plane along the
dotted lines.

ciL

n fi

_i 4-^ ^\

A^— J B c L — 1

^7 a

FIG. 54.

FIG. 55. — Perspective.

5. Ballast the model with a paper clip placed at the end of
the strip AB, and with another in the neighbourhood of L.

STABILITY IN STILL AIR 141

The exact positions are to be found by experiment, and it
may therefore be as well to turn the cardboard about its
glued end before the glue has set.

If this glider is thrown upwards towards the sky, it will
right itself and glide away in the attitude shown in fig. 55.
Now the centre of gravity of a glider of this kind lies some-
where about G.

On the other hand, the point sometimes termed the
" centre of lift " is situated on the plane at the spot which,
in equilibrium, is on the perpendicular from the centre of
gravity and shown at S. This point S lies below the centre
of gravity.

Now, if an aeroplane ought to be considered as suspended
in space from a so-called " centre of lift," its centre of
gravity could not, perforce, be anywhere but below this
" centre of lift."

In the case just mentioned the opposite took place, which
shows very clearly that this idea of a " centre of lift " is
erroneous.

An aeroplane has one centre only, its centre of gravity.

CHAPTER VIII
STABILITY IN STILL AIR

LATERAL STABILITY

FOR the complete solution of the problem of aviation the
aeroplane must possess, in addition to stable longitudinal
equilibrium, stable lateral equilibrium or, more briefly,
lateral stability.

The fundamental principle laid down hi Chapter VI. is
equally applicable to lateral equilibrium.*

But hi the case of longitudinal equilibrium the move-
ments that had to be considered hi respect of stability
could be simply reduced to turning movements about a
single axis, the pitching axis. The matter becomes exceed-
ingly complicated hi the case of lateral equilibrium, for
the turning movements can take place about an infinite
number of axes passing through the centre of gravity and
situated in the symmetrical plane of the machine.

For instance, assume that the aeroplane diagrammatically
shown hi fig. 56 were moving horizontally and that the
path of the centre of gravity G were along GX. If the
machine were to turn through a certain angle about the
path GX, clearly no change would take place in the manner
in which the air struck any part of the machine,f and
no turning moment would arise tending to bring the machine

* From the point of view of equilibrium and stability, the aeroplane
may be regarded as if it were suspended from its centre of gravity, and
were thus struck by the relative wind created by its own speed.

t Assuming, of course, that the turning movement does not alter the
path of the centre of gravity.

STABILITY IN STILL AIR 143

back to its former position or to cause it to depart there-
from still further.

It can therefore be stated that the lateral equilibrium
of an aeroplane is neutral about an axis coincident with
the path of the centre of gravity.

But when we come to consider turning movements about
other axes such as GXj or GX2 which do not coincide with
the path of the centre of gravity, it is evident that such
movements will have the effect of causing the aeroplane to
meet the air dissymmetrically, and consequently to set up
lateral moments tending to increase or diminish the tilt of
the machine — that is, upsetting or righting couples.

Before going further it is readily evident that, the axis

X—

FIG. 56.

GX being neutral, axes such as GXj and GX2, lying on
opposite sides of GX, will have a different effect, and that
a turning movement begun about one series of axes will
encounter a resistance due to the dissymmetrical reaction of
the air which it creates, while any turning movement begun
about the other series of axes, again owing to the dis-
symmetrical reaction of the air, will go on increasing until
the machine overturns.

The former series will be known as the stable axes, the
latter as the unstable axes. The neutral axis is that co-
inciding with the path of the centre of gravity. Further,
the term raised axis will be used to denote an axis with
its forward extremity raised like GXj and lowered axis
for that which, like GX2, has its forward extremity
lowered.

144 FLIGHT WITHOUT FORMULA

The shape of the aeroplane determines which axis is
unstable.

In many aeroplanes, and in monoplanes in particular,*
the forward edges of the wings do not form an exact straight
line, but a dihedral angle or V opening upwards.

We shall also have to examine — though the arrangement
hi question has never to the author's knowledge been
adopted in practice — the case of the machine with wings
forming an inverted dihedral or A-t Lastly, the forward
edges of the two wings may form a straight line, and such
wings will hereafter be described as straight wings.

In an aeroplane with straight wings, a turning movement
imparted about an axis situated in the symmetrical plane
of the machine increases the angle of incidence if the axis

FIG. 57.

is a lowered axis, and diminishes the angle if the axis is a
raised axis. This can easily be proved geometrically, and
can be shown very simply by the following experiment.

Make a diagonal cut in a cork, as shown in fig. 57 (front
and side views). In this cut insert the middle of one of
the longer sides of a visiting-card, and thrust a knitting-
needle or the blade of a knife into the centre of the cork
on the side where the card projects. Now place this con-

* In the case of large-span biplanes the flexing on the planes in flight
forces them into a curve which in its effects is equivalent, for purposes
of lateral stability, to a lateral dihedral.

t The " Tubavion " shown at the 1912 Salon is stated to have flown
with wings thus disposed.

TRANSLATOB'S NOTE. — The same device was adopted by Cody in his
earlier machines, and in the " June Bug," the first machine designed by
Glenn Curtiss.

STABILITY IN STILL AIR

145

trivance in the position shown in fig. 58, with the needle
horizontal and at eye-level. If the needle is rotated slowly,
the card will always appear to have the same breadth
whatever its position.

If this visiting-card is taken to represent the straight

FIG. 58.

wings of an aeroplane struck by the wind represented by
the line of sight, this shows that a turning movement about
the neutral axis of an aeroplane with straight wings
produces no change in the angle of incidence, as already
known.

FIG. 59.

But if the needle is inserted in the position shown in
fig. 59, it will be found that by rotating the needle without
altering its position, the breadth of the card will appear to
increase, thus showing, retaining the same illustration, that
when the axis of rotation of a machine with straight wings
is a lowered axis, the incidence increases as the result of
the turning movement.

146

FLIGHT WITHOUT FORMULA

This effect is the more pronounced the smaller the angle
of incidence.

But if the needle is inserted as shown in fig. 60, the
breadth of the card when the needle is rotated will appear
to diminish.

If the needle is parallel to the card, a turn of the needle
through 90° brings the card edge-on to the line of sight.

Lastly, if the needle and the card are in converging
positions, a slight turn of the needle brings the card edge-on,
and beyond that its upper surface alone is in view.

From this we may conclude that if the axis of rotation
of an aeroplane with straight wings is a raised axis, the
angle of incidence diminishes as the result of a turning

FIG.

movement, and if the axis is raised to a sufficient degree,
the angle of incidence may become zero and even negative.

This effect is the more pronounced the larger the angle of
incidence.

It should be noted that in neither case is the action
dissymmetrical and that both wings are always equally
affected. In other words, should a machine with straight
wings turn about an axis lying within its plane of symmetry,
no righting or upsetting couple is produced by the turning
movement.

On the other hand, if the eye looks down vertically upon
the cork from above, it will be seen that a turning move-
ment about a lowered axis has the effect of causing the
rising wing to advance, while in the case of a raised axis
a turning movement causes it to recede (fig. 61). Now, by

STABILITY IN STILL AIR

147

advancing a Aving, the centre of pressure is slightly shifted ;
this may produce a couple tending to raise the advancing
wing.

Should the advancing wing be the lower one, which
corresponds to the case of a raised axis, this couple is a
righting couple. In the reverse case it is an upsetting
couple.

In this respect, for aeroplanes with straight wings a
raised axis is stable, a lowered axis unstable.

This effect in itself is very slight, but it represents the
nature of the lateral equilibrium of an aeroplane with

Raised Axis.

FIG. 61.

straight wings ; for if it were not present, a machine with
straight wings would be hi neutral equilibrium and possess
no stability.

But as soon as the wings form a lateral dihedral, whether
upwards or downwards, this effect practically disappears
and becomes negligible. This is the case next to be
examined.

Let us suppose, to begin with, that the wings form an
upward lateral dihedral, or open V- Each of the wings
may be considered in the light of one-half of a set of straight
wings which has begun to turn about the axis represented
by the apex of the V> the movement of each whig being

148

FLIGHT WITHOUT FORMULAE

in the opposite direction, i.e. while one is falling the other

is rising.

The considerations set forth above show that a turning

movement about a raised axis causes the incidence of the

rising wing to diminish
while that of the fall-
ing wing increases ; the
contrary takes place in
the case of a lowered
axis.

This is easily demon-
strated by tilting up-

FI<J. 62.— Stable. Lateral V and raised axis. Ward the two halves of

the visiting-card used in

the previous experiment. If the contrivance is looked at
as before, so that the axis of the cork is horizontal and
on a level with the eye, it will be found that any rotation
about the needle, when this is directed upwards, causes the
rising wing to appear to
diminish in surface while
the falling wing in-
creases (see fig. 62).
But if the needle points
downwards, the opposite
takes place (fig. 63).

In the first case, there-
fore, the turning move-
ment produces a righting
couple, in the second case
an upsetting couple.

This effect is the more
pronounced the larger
the angle of incidence.

Therefore in the case of wings forming a lateral dihedral,
a raised axis is stable, a lowered axis unstable, and the
more so the greater the angle of incidence.

This effect is added to the secondary effect already referred

FIG. 63.— Unstable. Lateral V with lowered
axis.

STABILITY IN STILL AIR

149

to in the case of straight wings ; but as soon as the dihedral
is appreciable, the former effect becomes by far the stronger.
Now consider the case of wings forming an inverted
dihedral or A- The same line of reasoning shows (see figs.
64 and 65) that :

In the case of wings forming an inverted lateral dihedral
a raised axis is un-
stable, a lowered axis
stable, and this the more
so the smaller the angle
of incidence.

In this case the
secondary effect acts

in opposition, but it Fic" ^-^fsed wif6™1 Awith
becomes negligible as
soon as the inverted dihedral is appreciable.

These various effects are increasingly great, it will
be readily understood, as the span is increased in size,
for the upsetting or righting couples have lever arms
directly proportional to the span. Besides, but quite
apart from the value of the incidence in a given case,

it is clear that the
righting couples are
greater the higher the
speed of flight, since
they are proportional
to the square of the
speed. Broadly speak-
ing, therefore, though
with certain reserva-
tions into which we need not here enter in detail, it may
be stated that the higher the flying speed the greater is
lateral stability.

Although the stability or instability of any axis depends
chiefly on the main planes, other parts of the aeroplane can
affect it to a certain extent, hence their effect should be
taken into account as well.

150 FLIGHT WITHOUT FORMULAE

The tail plane, which is usually straight, only affects
lateral stability to an inappreciable extent.

But it should be noted, as already stated, that any turning
movement about an axis other than the neutral axis will
affect the incidence at which the tail plane meets the air ;
and, since such a turning movement also affects, as already
known, the incidence of the main plane, this dual effect
must needs disturb the longitudinal equilibrium of the
machine. Hence, we arrive at the general proposition that
rolling begets pitching.

As regards the remaining parts of the aeroplane — fuselage,
chassis, vertical surfaces, etc. — they experience from the
relative wind, when the aeroplane turns about an axis
in the median plane, certain reactions which may be dis-
symmetric and would thus affect the equilibrium of the
machine on its flight-path. More particularly when the
parts in question are excentric relatively to the turning
axis can they influence — though usually only to a small
extent — lateral equilibrium.

For the sake of convenience and in a manner similar to
that previously adopted in the case of the detrimental
surface, the effects of all these parts may be concentrated
and assumed to be replaced by the effect of a single fictitious
vertical surface, which may be termed the keel surface,
which would, as it were, be incorporated in the symmetrical
plane of the machine.

Certain parts of the aeroplane, such as the vertical rudder,
the sides of a covered-in fuselage, vertical fins, form actual
parts of the keel surface.

Evidently, according to whether the pressure exerted on
the keel surface, by reason of a turning movement about
a given axis, passes to one side or the other of this axis, the
couple set up will be either a righting or upsetting couple.

It is easily shown that a keel surface which is raised
relatively to the axis of rotation can be compared, pro-
portions remaining the same, to a plane with an upward
dihedral, or V, and that a keel surface which is low relatively

STABILITY IN STILL AIR

151

to the axis of rotation to a plane with a downward dihedral
or A-

For this purpose, the cork, visiting-card, and needle
previously employed may be discarded in favour of a
visiting-card fixed flag- wise to a knitting-needle. It is
clear, as shown in fig. 66, that when the axis of rotation
is raised, a high keel surface renders this axis stable and a
low keel surface renders it unstable, while the reverse is the
case if the axis of rotation is lowered (see fig. 67).

Unstable. Raised axis and low kt
FIG. 66.

But this effect, as previously explained, is of small
importance as compared with that due to the shape of the
main plane ; for, while the pressures on the keel surface are
never far removed from the axis of rotation, the differential
variations in the pressure exerted on the two wings of a
plane folded into a dihedral have, relatively to the axis, a
lever arm equal to half the span of the wing, and accord-
ingly these variations are considerable.

The effect of the dihedral of the main plane is therefore
not equivalent in magnitude to that of the keel surface
formed by the projected dihedral (fig. 68). The dihedral

152

FLIGHT WITHOUT FORMULA

has a much greater effect on lateral stability than a similar
keel surface.

We now know the position of the stable and the un-
stable axes of rotation according to the particular struc-

Unstable. Lowered axis and high keel.
\

Stable. Lowered axis and low keel.
FIG. 67.

ture of the aeroplane, and we have found that the same
machine can be stable laterally for one axis of rotation,
and unstable for another.

This is scarcely reassuring and inevitably leads to the

FIG. 68.

question : About which axis can an aeroplane, flying freely
in space, be brought to turn ?

In the first place, the position of the axes obviously
depends on the causes which can bring about the turning
movement. But these causes are known : so far as lateral

STABILITY IN STILL AIR

153

equilibrium is concerned, they can only consist in excess of
pressure on one wing or on the keel surface.

Here, then, we have one important element of the ques-
tion already settled. Nevertheless, the problem cannot
be solved in its entirety without having recourse to ordinary
mechanics and calculations, though the results thus obtained
may well be called into question, since the calculations have
to be based on hypotheses which are not always certain in
the present state of aerodynamical knowledge.

Without attempting to examine this difficult problem in
all its details, we may never-
theless remark that in its
solution the most important
part is played by the dis-
tribution of the masses
constituting the aeroplane
or, in other words, by its
structure considered from
the point of view of inertia.

Let us take a long iron
rod AB (fig. 69), ballasted
with a mass M, and sus-
pend it from its centre of
gravity G ; add a small
pair of very light wings in the neighbourhood of the centre
of gravity.

If, with a pair of bellows, pressure is created beneath one
of the wings, the device will start to oscillate laterally, and
these oscillations will obviously take place about the axis
of the iron bar. If this is placed in the position shown in
fig. 69, the axis of rotation will be a raised axis ; if in the
position illustrated in fig. 70, it will be a lowered axis.

Now every aeroplane, and every long body in fact, has a
certain axis passing through the centre of gravity, about
which axis we can assume the masses to be distributed, as
in the case of the present device they are about the axis of
the iron bar.

FIG. 69.

154

FLIGHT WITHOUT FORMULA

Lateral oscillations tend to take place about this axis,
which may be termed the rolling axis. The term, it is
true, is not absolutely accurate, and lateral oscillations do

not take place mathe-
matically about this
axis ; but at the same
time, as further investi-
gations would show, the
true rolling axes only
differ from it to a very
slight extent, and are
always slightly more

.-,,„ raised than the rolling

M ^rj{ FlG' 70' axis. This brings us to the

moment of rolling inertia.

In Chapter VII. was defined the moment of inertia of
a body about any axis ; in the examination of longitudinal
stability the moment of inertia of an aeroplane about its
pitching axis was considered as the moment of pitching
inertia. But in the present case we have only to deal
with the moment of inertia of an aeroplane about its rolling
axis — that is, its moment of rolling inertia.

As a matter of fact, the true axis of lateral oscillations
coincides more closely with the rolling axis as, on the one
hand, the incidence of the main plane is nearer to the lest
incidence (see Chapter III.) and the corresponding drag-to-
lift ratio is smaller, and, on the other hand, as the ratio
between the moment of rolling inertia and the moment
of pitching inertia is smaller.

Owing to the fact that this latter ratio is very small in
the diagrammatic case just considered, the lateral oscilla-
tions of this device take place almost exactly about the
rolling axis, i.e. about the axis of the iron bar.*

* The moment of rolling inertia is very slight, since those parts which
are at any distance from the rolling axis, i.e. the wings, are very light,
while the moment of pitching inertia is great, owing to the length and
the weight of the iron bar.

STABILITY IN STILL AIR 155

From all this it is clear that, according to the position of
the aeroplane in flight, its natural axis of lateral oscillation,
or approximately its rolling axis, will be either a raised or
a lowered axis. For an aeroplane to possess lateral stability,
its natural axes of oscillation must obviously be stable axes.

Thus, if the wings of an aeroplane form an upward
dihedral or y, or if the machine has a high keel surface, its
natural axes of oscillation must be raised axes, if lateral
stability is to be ensured. This condition is complied with
if the rolling axis of the aeroplane is itself a raised axis,
and even when the rolling axis is slightly lowered, since
the natural axes of oscillation are relatively slightly raised.

It is also clear that the stability will be better the greater
the angle of incidence.

On the other hand, if the main planes form a downward
dihedral or A> the machine will be unstable laterally if the
rolling axis is a raised one or even if it is only slightly
lowered. But the aeroplane can be made stable if its
rolling axis is lowered to a sufficient extent, and the more
so the smaller the angle of incidence.

This conclusion is distinctly interesting since it is directly
at variance with the views held by the late Captain Ferber,
whose great scientific attainments lent him all the force of
authority, to the effect that an upward dihedral was essential
to lateral stability.

But it is even more important by reason of the fact —
which will be duly discussed in the final chapter, — already
noted by Ferber himself, that whereas the upward dihedral
or V is disadvantageous in disturbed air, the downward
dihedral has distinct advantages in this respect.

On the whole, however, Ferber's view is correct at present,
since in the majority of aeroplanes of to-day the rolling
axis is practically identical with the trajectory of the
centre of gravity or only very slightly lowered. But in
an aeroplane with a rolling axis lowered to an appreciable
extent, the upward dihedral might be highly injurious from
the point of view of lateral stability, whereas the inverted

156 FLIGHT WITHOUT FORMULA

dihedral or A would, contrary to general opinion, be
eminently stable.

How is this arrangement to be carried out in practice ?

The rolling axis is a line which passes through the centre
of gravity and lies close to the masses situated at the end
of the fuselage, such as the tail plane and controlling
surfaces. When the centre of gravity is normal, this line
consequently lies along the axis of the fuselage. But if
the centre of gravity is situated low relatively to the wings,
the rolling axis is also lowered. The same would occur if
the machine was so arranged as to fly with its tail high, so
that the axis of its fuselage would form an angle, distinctly
greater than the normal incidence, with the chord of the
main plane.

On the other hand, a low centre of gravity, if unduly
exaggerated, presents certain disadvantages.

The best method of obtaining a rolling axis such that
the inverted dihedral of the main plane produces lateral
stability would seem to be by combining both devices, i.e.
by slightly lowering the centre of gravity and raising the
tail in flight.

This conclusion was formed by the author several years
ago ; and in 1909, somewhat fearful of running counter to
the authoritative views of Captain Ferber, the opinion was
sought of the eminent engineer, M. Rodolphe Soreau, another
recognised authority, in regard to the position of the axis
about which an aeroplane's natural lateral oscillations take
place. In 1910 in a previous work,* the author first enun-
ciated in definite form the proposition that an aeroplane
with its main planes arranged to form an inverted dihedral
could, under given conditions, remain stable laterally.
Since then the point has been dealt with in an article in
La Technique Aeronautique and in a paper read before the
Academic des Sciences.f

* The Mechanics of the Aeroplane (Longmans, Green & Co.).
f La Technique Aeronautique, December 15, 1910 ; Comptes Rendus,
May 15, 1911.

STABILITY IN STILL AIR 157

Summing up :

( 1 ) In aeroplanes of the shape hitherto generally employed

a straight plane produces no lateral stability, apart
from the very slight stabilising effect produced by
the secondary cause, already referred to.

(2) In such aeroplanes a dihedral angle of the wings or

the use of a high keel surface produces lateral
stability, and this in an increasing degree as the
angle of incidence is greater.

(3) If the centre of gravity of the aeroplane is low, or if

its tail hi normal flight is high (or if both these
features are incorporated in the machine), an
inverted dihedral or A of the wings with a low
keel surface may produce lateral stability, and this
to an increasing extent the smaller the angle of
incidence.

Lateral stability, therefore, depends on several different
parts of the structure, but it can never attain the same
magnitude as longitudinal stability, which is easily explained.
For, whereas in the case of longitudinal stability any
angular displacement in the sense of diving or stalling
affects to its full extent the angle of the main and the
tail planes, as regards lateral stability a great angular
displacement in the sense of rolling is required to pro-
duce even a slight difference in the incidence of the two
wings.

The righting couples are therefore much smaller in
the lateral than the longitudinal sense for any given
oscillations. If, as hi Chapter VI., the degree of lateral
stability of an aeroplane is represented by the length of
a pendulum arm, it will be found that even with the
most stable machines this length is hardly in excess of
0'5 m., while attaining 2-5 and even 3 m. in the case of
longitudinal stability.

As with longitudinal stability, so here again there exists
a condition of stability of oscillation — that is, a definite

158 FLIGHT WITHOUT FORMULA

proportion must exist between the stabilising effect of
the shape of the machine and the value of its moment of
rolling inertia, so that the lateral oscillations can never
increase to the point of making the aeroplane turn turtle.
For this reason, since lateral stability is relatively small,
the moment of rolling inertia should not be too great.
On the other hand, an increase in span, which increases
this moment of inertia, also gives the stabilising effect
a long lever arm. Hence, a middle course had best be
adopted.

Aeroplanes with a large rolling inertia oscillate slowly,
so that there is time to correct the oscillations, but these
tend to persist.

So far as the wind is concerned, it would appear an
advantage to concentrate the masses, thus keeping the
moment of inertia small (see Chapter X.).

A low centre of gravity, as already shown, increases to a
considerable extent the moment of inertia both to pitching
and to rolling. Hence, if unduly low, it may set up lateral
oscillations, which constitute the disadvantage previously
referred to. If, therefore, a low centre of gravity is resorted
to with the object of inclining the rolling axis to permit the
use of wings with an inverted dihedral or A, care 'should
be taken that it be not too low, and it would seem in every
respect preferable to obtain the same result by raising
the tail.*

Aeroplanes with little rolling inertia oscillate more
quickly than the others. If this is slightly disadvantageous
since these oscillations cannot be so easily corrected, quick
oscillations, on the other hand, possess the advantage of
being accompanied by a damping effect similar to that
existing in the case of longitudinal oscillations and referred
to above. For, if a plane oscillates laterally, the wind
strikes it at either a greater or smaller angle than when
it is motionless, and this becomes the more marked the
quicker the oscillation.

* Such a tail should obviously offer as little resistance as possible.

STABILITY IN STILL AIR 159

The small degree of lateral stability possessed by aero-
planes, especially of those with straight planes, would,
generally speaking, usually not suffice to prevent the
upsetting of the machine owing to atmospheric disturb-
ances, the more so since, as Chapter X. will show, the very
shapes and arrangements which produce lateral stability
may at times interfere with the flying qualities of the
machine in disturbed air.

Hence it is necessary to give the pilot a means of power-
ful control over lateral balance in order to counteract the
effects of air disturbances.

This means consists in warping, which was probably
first conceived by Mouillard, and first carried out in practice
by the brothers Wright. Other devices, such as ailerons,
have since been brought out, the object in each case being
to produce, differentially or not, an excess of pressure on
one wing.

The pilot therefore controls the lateral balance of his
machine, and this has to be constantly corrected and
maintained by him.

Naturally, the idea of providing some automatic device
to replace this controlling action by the pilot has arisen,
but this question will be left for discussion in the last
chapter, which deals with the effects of wind on the
aeroplane.

The rotation of a single propeller causes a reaction in
an aeroplane tending to tilt it laterally to some extent.
This could easily be corrected by slightlj7 overloading the
wing that shows the tendency to rise ; but in this event the
reverse effect would take place when the engine stopped,
either unintentionally or through the pilot's action when
about to begin a glide — that is, at the very moment when
longitudinal balance is already disturbed.

Lateral balance is bound to be disturbed in some degree
owing to the propeller ceasing to revolve, but it would seem
preferable that at the moment when this occurs both wings
should be evenly loaded.

160 FLIGHT WITHOUT FORMULAE

For this reason constructors generally leave it to the
pilot to correct the effect referred to by means of the warp
(which term includes all the different devices producing
lateral stability). Probably the effect is responsible for
the tendency which most aeroplanes possess of turning
more easily in one direction than the other.*

* Another effect due to the rotation of the propeller, the gyroscopic
effect, will be briefly considered in the following chapter.

CHAPTER IX
STABILITY IN STILL AIB

LATERAL STABILITY (concluded) — DIRECTIONAL
STABILITY — TURNING

OUR examination of lateral stability may well be brought
to a conclusion by considering the interesting lessons that
may be learned from experiments with little paper gliders.
First, we will take some of those gliders which have been
described in previous chapters and examine them in regard
to lateral stability.

Type 1 (see Chapter VII.). — This, it will be remembered,
is a simple rectangular piece of paper. It has already been
explained that it was necessary to bend it so as to form a
lateral V-

This arrangement is essential for obtaining lateral stability
with this particular glider, since its rolling axis, which
corresponds approximately with the fold along the centre,
is a raised axis, for the reason that the path followed by
the centre of gravity must be at a lesser angle than this
central fold, in order to give the gliders an angle of incidence.
Practice here will be seen to confirm theory.

Cut out a rectangular sheet of very stiff paper and, without
folding it, load it with ballast as shown in Chapter VII.

During the process of finding, by experiment, the correct
position for the ballast, it will be found that the flight of
such a glider is accompanied by considerable lateral
oscillations. More, these oscillations are both lateral and
directional ; in other words, the path followed by the

161

162 FLIGHT WITHOUT FORMULA

centre of gravity is a sinuous one, and the glider not only
tilts up on to one wing and the other in succession, but
each time it tends to change its course and swerve round
towards the lower whig, and thus it is virtually always
skidding or yawing sideways.

In this way it appears to oscillate not about an axis
passing through the centre of gravity, but about a higher
axis.

The reason for this, which will be entered into more fully
in connection with directional stability, is the extremely
small keel surface of such a glider. This might at first
appear to conflict with the fundamental principle,* but the
anomaly is simply due to the fact that the lateral oscillations
which, as always, do indeed occur about an axis passing
through the centre of gravity, are combined with the zig-
zag movement due to the small keel surface, which is
moreover the outcome of the oscillations.

The tendency to roll is the result of the very slight
lateral stability of a straight plane, which possesses practi-
cally no keel surface, and this tendency is counteracted by
nothing but the small secondary damping effect referred
to in Chapter VIII.

Oscillatory stability is therefore almost absent, and the
first rolling movement would increase until the glider was
overturned, but for the fact that, the air pressure being no
longer directed vertically upwards, the path followed by
the centre of gravity is deflected sideways and the glider
tends to turn bodily towards its down-tilted side.

The glider promptly obeys this tendency, for its mass is
feeble while it possesses practically no keel surface offer-
ing lateral resistance ; hence the centre of pressure moves
towards the side in which movement is taking place and
thus creates a righting couple. Consequently, in a measure,
the yawing saves the glider from overturning.

This tendency to yaw which is displayed by machines with

* That an aeroplane should be considered as being suspended from
its centre of gravity (see p. 93).

STABILITY IN STILL AIR 163

straight wings and a small keel surface is to be observed,
for it would seem to furnish the reason for the side-slips to
which aeroplanes devoid of a lateral dihedral are prone.

Now, if the sheet of paper is slightly folded upwards
from the centre, these various movements decrease, and,
finally, if folded up still further, disappear altogether.
The dihedral angle increases lateral stability and oscillatory
stability, while the considerable keel surface which the
glider now possesses stops all tendency to yaw.

Since the stabilising effect of the dihedral in the example
chosen is due to the rolling axis being a raised axis, it is
to be expected that, when launched upside down, the
glider will prove to be laterally unstable. This, in fact, is
what occurs if the dihedral is pronounced enough. The
glider immediately turns right side up.

If the glider has no dihedral, or only a slight one, and if
the span is reduced (in this case either the ballast must be
moved back or — and this is preferable — the wing-tips must
be turned up aft, for the weight of the ballast is now dis-
proportionate to the weight of the paper so that the centre
of gravity has moved forward), it may be observed that
the lateral oscillations become quicker, which is due to the
moment of rolling inertia having diminished.

Type, 2 (see Chapter VII.). — This model represents the
normal shape of an aeroplane (fig. 51). It has already
been explained that it should be given a slight dihedral,
which, in any case, it will tend to assume of its own accord
owing to the combined forces of gravity and air pressure.

This glider will be found to possess good lateral stability,
its rolling axis coinciding approximately with the central
fold, so that it is a raised axis.

Oscillatory stability is good and rolling almost absent.
If the glider is bent into a downward dihedral or A (this
fold should be somewhat emphasised in view of the tendency
of the glider to assume an ordinary V of its own accord),
it will overturn, which is quite in accord with theory, as
the natural rolling axis is a raised axis.

164 FLIGHT WITHOUT FORMULA

Type 3 (see Chapter VII.).— This is the " Canard " or tail-
first type (fig. 53). The wing-tips being bent vertically
upward, form a high keel surface, while, as in the previous
case, the glider naturally tends to assume a V-

Hence it is laterally stable, and the rolling axis, coincid-
ing with the central fold, is a raised axis.

There are no appreciable oscillations. If it is bent
downward into a f\, and this to a pronounced extent owing
to the stabilising effect of the fins, or if these latter are
folded down, the glider overturns.

These three types, therefore, are in accordance with the
principle laid down by Captain Ferber, to the effect that
a lateral V is necessary for lateral stability.

Examination of the following two models, on the other
hand, shows them to be in contradiction with this principle,
and bears out the author's contention that lateral stability
may be obtained in an aeroplane possessing a downward
dihedral or A-

Type 5* — 1. Cut out from a sheet of paper folded in two
the outline shown in fig. 71.

2. Cut away, hi the dimensions shown, the folded edge
at AB and CD.

3. Glue to the inside of the fold : (a) at AB, a small
strip of cardboard cut from a visiting-card (5 to 6 cm.
long and 1 cm. broad), (b) at CD a rectangular piece of
paper (4 cm. by 15 mm.).

4. Fold back the wings and the tail plane along the thick
dotted lines. The wings should be folded so as to form a
lateral A- (If any difficulty is experienced in maintaining
this shape, a thin strip of cardboard, 4 to 5 mm. wide,
may be glued along the forward edge.)

5. Affix the ballast, consisting of one or more paper
fasteners, to the end of the paper strip in front. (The
correct position must be found by experiment, and it
may be useful, for this purpose, to adjust the forward

* This number was given so as not to break the numerical sequence.
Type 4 was dealt with in Chapter VII.

STABILITY IN STILL AIR

165

strip of cardboard to the correct position before the glue
is quite dry.)

FIG. 71.

Type 6.—\. Cut out from a'sheet'of paper folded in two
the outline shown in fig. 72.

Fold along this line.

FIG. 72.

2. Fold back the wings and the tail plane along the
thick dotted lines. The wings are folded so as to form an

166 FLIGHT WITHOUT FORMULAE

inverted dihedral or A> the edge of the fold being upper-
most in this model.

3. Glue to the inside of the fold : (a) in front and so as
to form a continuation of the fold, a strip of cardboard cut
from a visiting-card, and measuring 5 or 6 cm. by 1 cm. ;
(b) in the rear and at right angles to the fold, another strip
of cardboard measuring 4 cm. by 15 mm.

4. Affix ballast in the shape of two or three paper clips
at the extremity of the foremost cardboard strip.

These models belonging to types 5 and 6 have to be
adjusted with great care and will probably turn over at
the first attempt, until balance is perfect, but this need not
discourage further attempts.

If they display a marked tendency to side-slip or yaw
or to turn to one side, the trailing edge of the opposite
whig-tip should be slightly turned up, until balance is
obtained.

But if the model rolls in too pronounced a fashion,
such oscillations may be caused to disappear either by
shifting the ballast or even by turning up or down the
rear edge of the tail plane. In some cases the same result
may be obtained simply by emphasising the A of the
wings.

Once they are properly adjusted, these gliders assume on
their flight-path the attitudes shown respectively in figs.
73 and 74. These are the attitudes imposed by the laws
of equilibrium ; and if the gliders are thrown skyward
anyhow, they will always resume these positions, provided
they are at a sufficient height above the ground.

Model 5 displays a slight tendency to roll, but model 6
follows its proper flight-path, which can be made perfectly
straight, in quite a remarkable manner.

This is all in accordance with the theory put forward in
Chapter VIII. Because the rolling axis is a lowered axis
— since in model 5 the centre of gravity is situated very low
and in model 6 the tail is very high — the inverted dihedral
or A of the wings produces stable lateral equilibrium.

STABILITY IN STILL AIR

167

On the other hand, in model 5 the centre of gravity is
exceptionally low ; hence the moment of rolling inertia is

FIG. 73.— Perspective.

PIG. 74. — Perspective.

great, so that oscillatory stability is not quite perfect and
there remains a tendency to swing laterally, whereas in

168 PLIGHT WITHOUT FORMULA

model 6 this defect is absent, since the centre of gravity is
only slightly lowered.

The latter arrangement is therefore the one to be adopted
in designing a full-size aeroplane of this type.

In both cases, by increasing the A the tendency to
oscillate is reduced, owing to the fact that, up to a certain
limit, the value of the righting couples is hereby increased
likewise. The same is true of a decrease in the angle of
incidence, effected either by displacing the ballast farther
forward or by adjusting the tail, because, as has been
shown previously, any decrease in the incidence augments
lateral stability if the wings are placed at a A-

Our theory would be even more conclusively proved
correct, if, when the wings were turned up into a V> the
glider overturned.

As a matter of fact this does occur sometimes, especially
with model 6, but not always, and with its wings so arranged
the model may still retain a certain amount of lateral
stability.

This apparent conflict of practice and theory may be
explained by the fact that, by turning up the wings of
such a model the centre of gravity is raised, since the
wings constitute an important part of the weight of these
little gliders ; consequently the rolling axis is also raised,
and since, as previously stated, lateral oscillation occurs
not precisely about the rolling axis but about a higher axis
still, the true rolling axis may prove to be a stable axis
for V-shaped wings. This is borne out further by the
fact that in many cases, and especially with model 6, this
does not occur and that the glider overturns.

With the kind assistance of M. Eiffel, the author carried
out hi the Eiffel laboratory a series of tests with a scale
model of greater size and so designed that its wings could
be altered to form either an upward or a downward dihedral,
and these tests appear to be conclusive.

The model, perfectly stable when its wings formed a
A > showed a strong tendency to overturn when the wings

STABILITY IN STILL AIR

169

formed a V- (The raising of the centre of gravity caused
by upturning the wings was neutralised by lowering the
ballast to a corresponding extent.)

But even if this further proof were absent, it would
nevertheless remain true — and the fact is most important,
as will be shown in Chapter X. — that it seems possible to
build aeroplanes, with wings forming an inverted dihedral
angle, which in spite of this are laterally stable.

DIRECTIONAL STABILITY

An aeroplane must possess more than longitudinal and
lateral stability ; it must maintain its direction of flight,
must always fly head to the relative wind, and must not
swing round owing to a slight disturbance from without.
This is expressed by the term directional stability.

In other words, an aeroplane should
behave, in the wind set up by its own
speed through the air, like a good *

weathercock.

In fig. 75, let AB represent, looking
downwards, a weathercock, turning about \ /

the vertical axis shown at 0, the direction
of the wind being shown by the arrow.

From our knowledge of the distribu-
tion of pressure on a flat plane (fig. 32,
Chapter VI.), it is clear that if the axis
O is situated behind the limit point of
the centre of pressure, the weathercock,
in order to be in equilibrium, would have
to be at an angle with the wind such
that the corresponding pressure passed
through the point 0. Hence, the weathercock would
assume the position A'B' or A*B".

It would be a bad weathercock because it formed an
angle with the wind. A good weathercock always lies
absolutely parallel with the wind, which thus always meets
it head-on.

A" A

FIG. 75.— Plan.

170 PLIGHT WITHOUT FORMULA

Therefore, in a good flat weathercock the axis of rotation
must be situated in front of the limit point of the centre
of pressure, i.e. in the first fourth from front to rear.

In so far as its direction in the air is concerned, an aero-
plane behaves in exactly the same way as the weather-
cock which we have termed its keel surface, the axis of
rotation being approximately a vertical axis passing through
the centre of gravity.

It can therefore be stated that for an aeroplane to possess
directional or weathercock stability, the limit point of the
centre of pressure on its keel surface, when it meets the air
at even smaller angles, must lie behind the centre of gravity.

Directional equilibrium is thus obviously stable, since
any change of direction sets up a righting couple, because
the pressure on the keel surface always passes behind the
centre of gravity.*

Directional stability is usually maintained by the means
already provided to secure lateral stability, the rear portion
of the fuselage, which is often covered in with fabric, con-
stituting the rear part of the keel surface. Moreover, this
is further increased by the presence of a vertical rudder
still further aft.

But there are certain machines in which special means
have to be taken to secure directional stability — the tail-
first or " Canard " machine is of this type.

In Chapters VI. and VII. it was stated that the fact that
this type of machine has its tail plane in front tends to
longitudinal instability, which is only overcome by the
unusually high stabilising efficiency of the main planes,

* Reference to Chapters VI. and VII. will show that longitudinal
equilibrium is also, in effect, weathercock equilibrium. But in this
respect, the planes must always form an angle with the relative wind,
which constitutes the angle of incidence and produces the lift. In
regard to longitudinal stability, the aeroplane should therefore be a bad
weathercock. Further, it will be shown in Chapter X. that, in con-
sidering the effect of the wind on an aeroplane, two classes of bad weather-
cocks have to be distinguished, and that an aeroplane should be, if the
term be allowed to pass, a " good variety of bad weathercocks."

STABILITY IN STILL AIR 171

which are of relatively great size and situated at a con-
siderable distance behind the centre of gravity.

The same is true in regard to directional stability, and
the existence in the forward part of the machine of a long
fuselage, comparable to a weathercock turned the wrong
way round, would speedily cause the aeroplane to turn
completely round if it were not provided with considerable
keel surface behind the centre of gravity. The necessity
for this arrangement will readily appear if, in the little
paper glider No. 3, already described, the vertical fins at
the wing-tips are removed. The glider will then turn
about itself without having any fixed flight-path.*

In Chapter VIII. it was shown that lateral stability
is affected by raising or lowering the vertical keel surrace.
But even if it is neither high nor low, and though it may
appear to affect only directional stability, every bit of keel
surface plays an important part in lateral stability. For
these two varieties of stability are not absolutely distinct.
Both, in fact, relate to the rotation of the aeroplane about
axes situated in the plane of symmetry.

When these axes are close to the flight-path of the centre
of gravity, only lateral stability comes in question ; but
when they are more nearly vertical, the rotary movement
about them belongs to directional stability.

Nevertheless, any turning movement about any axis
other than that formed by the path of the centre of gravity
plays its part in both lateral and directional stability, and
it is only in so far as it affects the one more than the other
that it is classified as belonging to lateral or directional
stability. The line of cleavage between these two varieties
of stability is by no means clear.

From this it follows, in the author's opinion, that the
means for obtaining lateral stability gain considerably in
effectiveness if they also produce directional stability. If

* To obtain good directional stability, those paper models with a
ballasted strip of cardboard in front were all provided with a vertical
fin in the rear.

172 FLIGHT WITHOUT FORMULA

aeroplanes with wings forming a A are eyer built, they
should be provided with a considerable amount of keel
surface aft (placed low rather than high).

In conclusion, it may be said that, of the three varieties
of stability, directional stability is at the present time the
most perfect, which is to be accounted for on the ground
that the pressure on the keel surface must always pass
behind the centre of gravity, whence arise strong righting
couples.

In the order of their effectiveness at the present day,
the three classes of stability can therefore be arranged as
follows : —

Directional stability.

Longitudinal stability.

Lateral stability.

By careful observation of the oscillations of an aeroplane
the truth of this statement will be borne out. Every
aeroplane betrays some tendency to roll ; at times it also
tends to pitch, but it hardly ever swerves from side to side
on its flight-path, zigzag fashion.

TURNING

The vertical flight-path of an aeroplane is controlled by
the elevator ; but the pilot must also be able to change his
direction and to execute turning movements to right and
left.

A few points of elementary mechanics may here be
usefully recalled.

If a body is freely abandoned to its own devices after
having been launched at a certain speed (omitting from
consideration the action of gravity), it continues by reason
of its inertia to advance in a straight line at its original
speed, and an outside force is required in order to modify
this speed or to alter the direction followed by the body.

A body following a curved path therefore only does so
through the action of an outside force.

If the body follows a circular path, the force which pre-

STABILITY IN STILL AIR 173

vents it from getting further away from the centre of the
circle, although its inertia seeks to propel it in a straight
line, to move away at a tangent, is termed centripetal force.

For instance, if a stone attached to the end of a string
is whirled round, it describes a circle instead of following
a straight line only because the string resists and exerts
on it a centripetal force. If this force is stopped and the
string is let go, the stone will fly off at a tangent.

On the other hand, a body, in this case the stone, always
tends to fly off ; it thus reacts, exerting in its turn on the
cause which maintains it in a circular path — in this case,
on the string — a force termed centrifugal force, which, in
accordance with the well-known principle of mechanics
concerning the equality of action and reaction, is exactly
equal and opposite to the centripetal force which causes it.

In the example chosen, the value of the centripetal and
centrifugal forces (the same in both cases) could be
measured by attaching a spring balance to the string. It
would be found that, as is easily shown in theory, this
value is proportional to the square of the speed of rotation
and inversely proportional to the radius of the circle
described.

From this it is clear that in order to curve the flight-
path of an aeroplane, that is, to make it turn, it is necessary
to exert upon it by some means or other a centripetal
force directed from the side hi which the turn is to be
made. This can be done by creating, through movable
controlling surfaces, a certain lack of symmetry in the
shape of the aeroplane which will result in a correspond-
ing lack of symmetry in the reactions of the air upon it.

The most obvious proceeding is to provide the aeroplane
with the same device by which ships are steered and to
equip it with a rudder. But, just as a ship without a
keel responds only in a slight measure to the action of a
rudder, so an aeroplane offering little lateral resistance —
that is, having but little keel surface — only responds to the
rudder in a minor degree.

174

FLIGHT WITHOUT FORMULAE

In order to make this clear, we will take the case of an
aeroplane entirely devoid of keel surface, though this is
an impossibility on a par with the case of an aeroplane
wholly devoid of detrimental surface, since the structure
of an aeroplane must perforce always offer some lateral
resistance, even though the constructor has tried to reduce
this to vanishing-point.

However, let us assume that such an aeroplane, having

FIG. 76.

its centre of gravity at G (fig. 76), is provided with a
rudder CD.

If the rudder is moved to the position CD', the aero-
plane will turn about its centre of gravity until the rudder
lies parallel with the wind. But there will not be exerted
on the centre of gravity any unsymmetrical reaction, any
centripetal force capable of curving the flight-path.

The aeroplane will therefore still proceed in a straight
line, and the only effect of the displacement of the rudder

STABILITY IN STILL AIR

175

will be to make the aeroplane advance crabwise, without
any tendency to turn on its flight-path.

But if the machine is equipped with a keel surface AB
(fig. 77), directional equilibrium necessitates that this keel
surface should present an angle to the wind, and become
thereby subjected to a pressure Q, whose couple relatively

Fro. 77.— Plan.

to the centre of gravity is equal and opposite to the pressure
q exerted 011 the displaced rudder CD'. Since Q is con-
siderably greater than q, there is exerted on the centre of
gravity, as the result of their simultaneous effect, a resultant
pressure approximately equal to their difference (which
could be found by compounding the forces), and this forms
a centripetal reaction capable of curving the flight-path —
that is, of making the machine turn.

176

FLIGHT WITHOUT FORMULA

It should be observed that the nearer the keel surface
is to the centre of gravity the greater is the centripetal
force set up by the action of the rudder. Similarly, the
intensity of this force also depends on the extent of the
keel surface. And lastly, since the centripetal force has
a value equal to the difference between the pressures Q
and q, it becomes greater the smaller the latter pressure.
Hence there is an advantage in using a small rudder, which

must, in consequence,
have a long lever arm
in order to balance the
effect of the keel surface.
A turn might also be
effected by lowering a
flap CD, as shown in
fig. 78 at the extremity
of one wing, this flap
constituting a brake.
In this case, too, a keel
surface is essential and
equilibrium would exist
if the couples set up by
the pressures Q and q,
exerted on the keel sur-
face and on the brake
respectively, were equal.
A centripetal reaction,
the resultant of these pressures, would act on the centre
of gravity and bring about a turn.

There remains a third and last means of making an
aeroplane perform a turn, and this requires no keel surface.
This consists in causing the aeroplane to assume a permanent
lateral tilt.

The pressure exerted on the plane (which is roughly
equal to the weight of the machine) is tilted with the aero-
plane and has a component p (fig. 79) which assumes the
part of centripetal force, and makes the machine turn.

FIG. 78.— Plan.

STABILITY IN STILL AIR

177

The machine can be tilted in various ways — for instance,
by overloading one of the wings. But the more usual
method is that of the warp, which has already been referred
to as the pilot's means of maintaining lateral balance.

By increasing the incidence, or its equivalent the lift, of
one wing-tip and decreasing that of the other, the former
wing is raised and the latter lowered, so that the machine
is tilted in the manner required to make a turn.

But in warping, the wing with increased lift also has
an increased drag or head-resistance, while the reverse
takes place with the other wing.

This secondary effect
is analogous with that of
the air brake just con-
sidered and is exerted in
the opposite way to that
required to perform the
turn. It is usually smaller
than the main effect of the
warp, but still interferes
with its efficacity. On
the other hand, in some
aeroplanes it may gain FIG. 79.— Front elevation,

the upper hand, as in the noteworthy case of the Wright
machines.

In order to overcome this defect, the brothers Wright
produced, through the means of the rudder (which played
no other part), a couple opposed to the braking effect,
which left its entire efficiency to the differential pressure
variation exerted on the wings by the action of the warp.
Further, the warp and rudder could be so interconnected
as to act simultaneously by the movement of a single
lever (this constituted the main principle of the Wright
patents).

This detrimental secondary effect could, it would appear,
be easily overcome by using a plane with wing-tips uptilted
in the rear as at BC in fig. 80.

Centripetal Force.

178 FLIGHT WITHOUT FORMULA

By depressing the trailing edge BC of the wings, which
are purposely made flexible, the lift is increased and the
drag diminished at the wing-tip. By turning up the trailing
edge the lift is decreased and the drag increased. Both
effects therefore combine to assist in making the turn
instead of impeding it. Instead, finally, of adopting this
particular warping method, the same result could be obtained
by using negative-angle ailerons.

It should be noted — and the fact is of importance both
so far as turning and lateral balance are concerned — that
the effect of the warp is definitely limited. It is known
that beyond a certain incidence (usually in the neighbour-
hood of 15° to 20°) the lift of a plane diminishes while the
drag increases rapidly.

If the warp is therefore used to an exaggerated extent,
the detrimental secondary effect referred to above comes

into play, with the result

» """'""'-— -^.^^^ ., that its effect is the reverse
•"• -Q G of the usual one. This may

prove a source of danger,
FIG. 80. -Profile. and jt might be weU ^

certain machines, if not to limit the warp absolutely, at
any rate to provide some means of warning the pilot that
he is approaching the danger-point.

Since the rudder sets up a couple tending to counteract
this secondary effect, it should be resorted to in case an
undue degree of warp causes a reverse action to the one
intended.

The banking of the planes which, as already seen,
may provoke a turn, always results from it ; for, as the
aeroplane swings round, the outer wing travels faster than
the inner wing, so that the pressure on the one differs
from that on the other, with the result that the outer one
is raised.

Therefore, if the centripetal force which causes the
turn does not originate from the intentional banking of
the planes, this banking which results from the turning

STABILITY IN STILL AIR 179

movement produces the necessary force to balance the
centrifugal force set up by the circular motion of the
machine.

It follows that the amount of the bank during a turn
depends on those factors which determine the amount of
centrifugal force. Hence, the bank is steeper the faster the
flying speed (being proportional to the square of the speed),
and the sharper the turn. It may therefore be dangerous
to turn too sharply at high speeds.

Equilibrium between centripetal and centrifugal force is
important simply in so far as it concerns the movement of
the aeroplane along its curved path, or, in other words, the
movement of its centre of gravity. But, in addition, the
machine itself should be in equilibrium about its centre of
gravity — that is, the couples exerted upon it by the air in
its dissymmetrical position during the turn must exactly
balance one another.

This position of equilibrium during a turn evidently
depends on various factors, among which are the means
whereby the turn has been produced and the distribution
of the masses of the machine.

For instance, if the turn is caused by banking it might
be thought that so long as the cause remained, the bank
would continue to grow more and more steep. But usually
this is not the case, for if the aeroplane possesses any
natural stability, the bank will itself set up a righting
couple balancing the couple which produced the bank.

The value of this righting couple depends, of course, on
the shape of the aeroplane and especially on the position
of its rolling axis. If the machine has little natural
stability, the pilot may have to use his controls in order to
limit the bank, as otherwise the machine would bank ever
more steeply and the turn become ever sharper until the
aeroplane fell.*

* Pilots have often mentioned an impression of being drawn towards
the centre when turning sharply.

180 FLIGHT WITHOUT FORMULA

As a rule, the warp is not used for producing a turn, for
the majority of machines possess sufficient keel surface to
answer the rudder perfectly.

Often the rudder aids the warp in maintaining lateral
balance : for instance, by turning to the left a downward tilt
of the right wing may be overcome.

Possibly in future the warp will become even less im-
portant, so that this device, which is generally thought
to have been imitated from birds (which have no vertical
rudder), may eventually vanish altogether.* The Paulhan-
Tatin " Torpille," referred to in previous chapters, had no
warp, neither had the old Voisin biplane, one of the first
aeroplanes that ever flew. This was due to the fact that
in both cases the keel surface (a pronounced curved dihedral
in the " Torpille," and curtains in the Voisin) was sufficient
to render the rudder highly effective.

It is to be noted that, whatever the cause of the turn, the
dissymmetrical attitude adopted as a result by the aeroplane
simultaneously causes the drag to increase while the lift
decreases owing to the bank. At the same time, the angle
of incidence alters, since any alteration in lateral balance
brings about an alteration in longitudinal balance, for rolling
produces pitching.

For these reasons an aeroplane descends during a turn.

The pilot feels that he is losing air-speed and puts the
elevator down. Theory, on the other hand, would appear to
teach that he ought to climb. But, as already stated, this
apparent divergence is due to the fact that theory applies
chiefly to a machine in normal flight. When an aeroplane
changes its flight and passes from one position to another,
effects of inertia may arise during the transition stage
which may vitiate purely theoretical conclusions, and in

* Although the author has carefully studied the flight of large soaring
and gliding birds in a wind, he has never found them to warp their wing-
tips to a perceptible extent to obtain lateral balance, while, on the other
hand, probably for this very purpose, they continually twist their tails
to right and left.

STABILITY IN STILL AIR 181

such a case theory must give way to practice. In any
event, practice need not necessarily remain the same should
the shape of the aeroplane undergo considerable alterations
and, more especially, if in future the lift coefficient becomes
very small.*

In conclusion, something remains to be said of the
gyroscopic effect of the propeller. Any body turning about
a symmetrical axis tends, for reasons of inertia, to preserve
its original movement of rotation.

The direction of the axis about which turning takes
place remains fixed in space, and, in order to alter it, a force
must be applied to it, which must be the greater the higher
the speed of rotation, the greater the movement of inertia,
and the sharper the effort to alter it.

But now arises the curious fact that if it is sought to
move the axis in a given direction, it will actually move in
a direction at right angles to this. This characteristic of
rotating bodies may be observed in the case of gyroscopic
tops, which only remain in equilibrium and only adopt a
slow conical motion when their axis becomes inclined
towards the end of their spinning, for this very reason.

Now a propeller which has a high moment of inertia,
especially if of large diameter, and turning at a great speed,
constitutes a powerful gyroscope (which is further increased
if the motor is of the rotary type).

It follows that any sudden action tending to modify the
direction of flight results in a movement at right angles to
that desired. Thus, a sudden swerve to one side may pro-

* It may be added that at very high speeds an aeroplane during a
sharp turn actually rises instead of coming down, but this is due to
quite a different cause. At the moment of turning, when already banked
and the rudder is brought into play, the machine for a fraction of time,
owing to its inertia, slides outward and upward on its planes. This
effect was particularly noticeable during the Gordon-Bennett race in
1913, when, long before the turning-point was reached, the aeroplanes
were gradually banked over, until at the last moment a sudden move-
ment of the rudder bar sent them skimming round, the while shooting
sharply upward and outward. — TRANSLATOR.

182 FLIGHT WITHOUT FORMULA

duce a tendency either to dive or to stall, according to which
side the swerve is made and to the direction of rotation of
the propeller.

Accidents have sometimes been ascribed to this gyroscopic
effect, but its importance would appear to have been greatly
exaggerated, and so long as the controls are not moved
very sharply, it remains almost inappreciable.

CHAPTER X
THE EFFECT OF WIND ON AEROPLANES

EVERY previous chapter related to the flight of an aero-
plane in perfectly still air. To round off our treatise, the
behaviour of the aeroplane must be examined in disturbed
air — in other words, we now have to deal with the effect of
wind on an aeroplane.

The atmosphere is never absolutely at rest ; there is
always a certain amount of wind. The two ever-present
characteristic features of a wind are its direction and its
speed. No wind is ever regular. Both its velocity and its
direction constantly vary and, save in a hurricane, these
variations do not depart from the mean beyond certain
limits. Hence, the wind as it exists in Nature may be
regarded as a normal wind, as if it had a mean speed and
direction, with variations therefrom.

These variations may be in themselves irregular or
regular up to a point. Near the ground the wind follows
the contour of the earth, encounters obstacles, and flows
past them in eddies ; hence it is perforce irregular, like the
flow of a stream along the banks.

Eddies are formed in the air, as in water : valleys, forests,
damp meadows where humidity is present — all these
produce in the air that lies above them descending currents,
sometimes called " holes in the air " ; while hills and bare
ground radiating the sun's heat produce rising currents
of air.

These effects are only felt up to a certain height in the
atmosphere, and the higher one flies the more regular

184 FLIGHT WITHOUT FORMULA

becomes the wind. In the upper reaches the wind seems
to pulsate and to undulate in waves comparable to the
waves of the sea.

The regular mean wind which reigns there may there-
fore be considered as possessing atmospheric pulsations,
propagated at a speed differing from the speed of the wind
itself, comparable to the ripples produced by throwing a
stone in flowing water — ripples which move at a speed
differing from that of the current itself.

This comparison of a regular wind with a flowing stream
enables the effect of such a wind on an aeroplane to be
studied in a very simple manner.

For the last time we will refer to that elementary principle
of mechanics applicable to any body moving through a
medium which itself is in motion — the principle of the
composition of speeds.

A speed, just as a force, may be represented by an arrow
of a length proportional to the speed and pointing in the
direction of movement.

For example, let us suppose that a boat is moving
through calm water at a speed represented by the arrow
OA (fig. 81).

Now, if instead of being still, the water were flowing at
a speed represented by the arrow OB, the ship, although
still heading in the same direction, would have a real speed
and direction represented by the arrow 00. This speed is
the resultant of the speeds OA and OB, and this composition
of speeds, it will be seen, is simply effected by drawing the
parallelogram.

The ship will appear still to be following the course OA,
which will be its apparent course, while in fact following
the real course 00.

Instead of a ship through flowing water, let us now take
the case of an airship or aeroplane moving through a current
of air or regular wind. Such a craft, while driven for-
ward through the air by its own motive power at the speed
it would attain if the air were perfectly calm, is at the

THE EFFECT OF WIND ON AEROPLANES 185

same time drawn along by the wind together with 'the
surrounding air, of which it forms, as it were, a part, and
this without the pilot being able to perceive this motion,
unless he looks at some fixed landmark on the ground.

An aeroplane may be likened to a fly in a railway carriage,
which is unable to perceive, and remains unaffected by,
the speed at which the train is moving.

In a free spherical balloon drifting before a regular wind
not a breath of air is perceptible. On board an aeroplane

FIG. 81.

or airship only the relative wind is felt which is created by
the speed of flight, no matter whether in still air or in wind.

In a side-wind, in order to attain to a given spot, a pilot
does not steer straight for his objective, but allows for the
drift, like a boatman crossing a swift-flowing river.

When the direction of the wind coincides with the path
of flight the speeds are either added to or subtracted from
one another ; for instance, an aeroplane with a flying speed
of 80 km. per hour in a calm will only have a real speed of
50 km. per hour against a 30-km. per hour wind, but will
attain 110 km. per hour when flying before it.

In order to be dirigible, an aircraft must have a speed

186 FLIGHT WITHOUT FORMULA

greater than that of the wind. In practice an aeroplane
virtually never flies in a wind of greater velocity than its
own flying speed, and hence is always dirigible.

The wind further affects the gliding path of an aeroplane.
For example, if an aeroplane with a normal gliding path
OA in a calm (fig. 82) comes down against the wind, its
real gliding path will be OC15 which is steeper than OA,
while with the wind behind it will be flatter, as shown by
OC2. The arrows OC^ and OC2 represent the resultant
speeds of the gliding speed OA in calm air and of the speeds
of the wind OBX and OB2.

But in all these different gliding paths, the gliding angle

of the aeroplane remains the same, since the apparent gliding
path relatively to the wind always remains the same.

If the speed of the wind is equal to that of the aeroplane,
the machine, still preserving its normal gliding angle,
would come down vertically and would alight gently on
the earth without rolling forward.

Birds often soar in this manner without any perceptible
forward movement, but, apart perhaps from the brothers
Wright during the course of their gliding experiments in
1911, no aeroplane pilot would appear to have attempted
the feat hitherto.*

* This statement is no longer correct. Many pilots have undoubtedly
flown in winds equal and even superior to their own flying speed.
Moreover, this vertical descent is sometimes made intentionally with
such machines as the Maurice Farman, the engine being stopped and
the aeroplane being purposely stalled until forward motion appears
to cease and the machine seems to float motionless in the air. —
TBANSLATOB.

THE EFFECT OF WIND ON AEROPLANES 187

A regular wind may be a rising current. In this case,
if sufficiently strong, it may render the gliding path hori-
zontal. Thus, if an aeroplane in calm air glides at a speed
OA (fig. 83), which has a horizontal component equal
to 15 m. per second, and follows a descending path of 1
in 6, a regular ascending current with a speed OBX or OB2,
with a vertical component equal to 2-5 m. per sec., would
enable an aeroplane to glide horizontally.

The existence of such ascending currents is sometimes
taken in order to explain the soaring flight practised by
certain species of large birds over the great spaces of the
ocean or the desert. But it is difficult to accept this as
the only explanation of this wonderful mode of flight,

which often extends for hours at a time, and would pre-
suppose the permanency of such rising currents. Another
explanation will be given hereafter.

We may now examine the effects on an aeroplane of
irregularities in the wind.

Any disturbance in the air may at any time be character-
ised by the modification in speed and direction of the wind ;
such modifications could be measured by means of a very
sensitive anemometer mounted on a universal joint.

The first effect of a disturbance is to tend to impart its
own momentary speed and direction to anything borne
by the air which it affects. Very light objects, feathers,
tissue-paper, etc., immediately yield to a gust.

If an aeroplane were devoid of mass, and therefore of
inertia, it would behave in the same way ; it would instantly
assume the new speed and direction of the wind and would

188 FLIGHT WITHOUT FORMULA

promptly obey its every whim. In this case the pilot
would be unable to perceive, except by looking at the
ground, any gusts or their effect ; for him it would be the
same as though he were flying in a regular wind.

But all aircraft possess considerable mass, and therefore
do not immediately obey the modifications resulting from
a wind gust in which they are flying. The disturbance
therefore exerts upon it, during a variable period, a certain
action, also variable, which can be likened to that which
would be experienced if the movements of the aeroplane
were restrained. This action, which may be termed the
relative action of a disturbance, modifies both in speed and
in direction the relative wind which the aeroplane normally
encounters, and these modifications can be felt by the
pilot and measured by an anemometer.

For the sake of simplicity, let us suppose that a wind of
a certain definite value is quite instantaneously succeeded
by a wind of another value, the wind being regular in each
case. A craft without mass would forthwith conform to
the new wind. The primary gust effect would be complete,
its relative action would be zero.

For any craft possessing mass the primary gust effect
would at first be zero and the relative action at a maximum ;
but, as the machine gradually yields to the gust, the rela-
tive action grows smaller and finally vanishes altogether
when the aeroplane has completely conformed to the new
wind. The greater the inertia of the machine, the longer
will be the transition period.

Still keeping to our hypothesis of an instantaneous
change of condition, an anemometer fixed in space and
another carried on the aeroplane might for one brief instant
record the same indications ; but while those of the fixed
anemometer would be constant, the other instrument
would sooner or later, according to the aeroplane's inertia,
return to its original indications.

If it is remembered that gusts, even the most violent,
are never perfectly instantaneous, it seems probable that

THE EFFECT OF WIND ON AEROPLANES 189

the relative action of a gust on an aeroplane is never so
intense as it would be were the machine fixed in space,
and that it dies away the more quickly the lighter the
aeroplane.

But the pilot of a machine in flight does not perceive
this relative action in the same way that he would if the
machine were immovable — for instance, if the aeroplane
were struck by a gust coming from the right at right angles,
the pilot of a stationary aeroplane would only feel the gust
on his right cheek, while in flight he would only perceive
the existence of a gust by the fact that the relative wind
was just a little stronger on his right cheek than on the
left. It is simply a question of the composition of

We have distinguished a primary gust effect and a rela-
tive effect. The results of each may now be examined.

The primary effect modifies in magnitude and in direction
the real speed of the aeroplane, which yields the more
slowly the greater its mass and inertia.

Now, instead of consisting, as our hypothesis required,
of an instantaneous succession of two winds of different
value, a gust is a more or less gradual and wavelike modi-
fication of the mean speed of the wind, lasting usually not
more than a few seconds.

Hence, if the aeroplane's inertia be sufficient, the cause
may cease before the gust has exerted its primary effect on
the aeroplane, the whole energies of the gust being absorbed
in producing the relative effect.

The direction of flight and the real speed of the aeroplane,
provided it has enough inertia, may consequently be only
slightly altered by the gust which would pass like a wave
past a floating body. This is why, whereas a toy balloon
is tossed by every little gust, a great passenger balloon sails
majestically on its way without being affected in the
slightest degree.

Why, therefore, should this not be the case with an
aeroplane which has a mass not differing widely from that

190 FLIGHT WITHOUT FORMULA

of a balloon ? The cause must be sought for in the relative
effect of the gust. This relative effect is only slight in the
case of a balloon which is based on static support according
to the Archimedean law ; but it affects the very essence of
the equilibrium of an aeroplane based on the dynamic
principle of sustentation by its speed and incidence.

Any variation in the speed or direction of the relative
wind, therefore, usually affects the value of the pressures
on the various planes, and consequently further affects its
attitude hi the air which is determined by a perpetual
equilibrium.

The effects produced by the relative action of a gust may
be divided into two classes : the displacement effect and the
rotary effect.

The displacement effect is that produced by the relative
action of the gust on the machine as a whole, and seen
in the modification of the path followed before by the
centre of gravity and the speed at which it moved until
then.

The displacement effect must not be confused with the
primary gust effect previously referred to.

For instance, if an aeroplane in horizontal flight is struck
head-on by a horizontal gust, the primary gust effect takes
the shape of a reduction in the real flying speed, which
reduction is the greater the smaller the inertia of the
machine. But this will not alter the horizontal nature of
the flight-path.

On the other hand, the displacement effect produced by
the gust will result in raising the whole machine which,
owing to its inertia and in increasing measure as its inertia
is greater, experiences an increase in the speed of the
relative wind, with the result that the lift on the planes
also increases.

The rotary effect is that produced by the relative action
of the gust on the equilibrium of the aeroplane about its
centre of gravity. This is due to the fact that the modi-
fications in the relative wind destroy the harmony between

THE EFFECT OF WIND ON AEROPLANES 191

the pressures on the various parts of the aeroplane, which
balanced one another and thus maintained the machine in
stable equilibrium.

Certain rotary effects are due to the fact that no gust is
instantaneous, but always moves at a speed which, however
great, is still limited. A gust may therefore first strike
one part of the aeroplane and produce a first rupture of
equilibrium ; then, continuing, it may strike the opposite
side which may already have been shifted out of position,
and affect this in turn either in the sense of restoring equili-
brium or the reverse.

The displacement and rotary effects due to a gust will
now be successively examined, beginning with those which
affect equilibrium of sustentation and longitudinal equili-
brium, these being closely interconnected. For the time
being, therefore, we will only deal with gusts moving in
the plane of symmetry of the aeroplane — that is, with
straight gusts, which affect the speed and the angle at
which the relative wind meets the planes.

First, let us examine the displacement effect. It will
result in a modification in the lift of the planes. The lift,
normally equal to the weight of the machine, has for its
value the lift coefficient of the planes multiplied by their
area and the square of the speed. If the lift coefficient
remains constant, and the relative wind increases as a
result of the gust, the lift of the planes increases ; if the
speed of the wind diminishes, so does the lift.

It is readily seen that in the case of small variations in
the speed, the variations in the lift are increasingly large,
the greater the weight of the machine and the lower its
normal flying speed. These variations depend neither on
the wing area nor on the value of the lift coefficient.

For instance, if an aeroplane weighing 400 kg. and flying
at 20 m. per second or 72 km. per hour, experienced, as the
result of a gust from the rear, a decrease in the relative
speed of 2 m. per second, the lift will decrease by 76 kg. If
it weighed 600 kg. instead of 400, its normal flying speed

192 FLIGHT WITHOUT FORMULA

being still 20 m. per second, the same decrease in the speed

600
would bring about a reduction in the lift of 76 x = 114 kg.

proportional to the weight.

If, weighing 400 kg., its normal speed were 30 m. per
second instead of 20, the same decrease of 2 m. per second in
the speed would produce a reduction in the lift of only
52 kg. instead of 76 as before.

These results remain true irrespectively of the plane
area and the lift coefficient.*

Now, suppose that, the speed of the relative wind re-
maining constant, the angle at which it meets the aero-
plane changes ; the value of the angle of incidence of the
planes is thereby modified and with it the lift coefficient.
The lift therefore also varies in this case, and a simple cal-
culation shows that these variations are the greater the
greater the weight and the smaller the lift coefficient.

For example, a machine weighing 400 kg. and possessing
a lift coefficient of 0-05, will, if this lift coefficient is re-
duced by 0-005 — which is equivalent to lessening the
angle of incidence by one degree — experience a loss of lift of
about 40 kg. If the weight were 600 kg., the loss of lift
would be 60 kg.

If it weighed 400 kg. and the normal lift coefficient were
0-025 instead of 0-05, the loss of lift resulting from a re-

* The method of calculation is quite simple.

Example.— it the weight is 400 kg. and the speed 20 m. per second —
the square of the latter being 400, — the product of the plane area and
the lift coefficient remains 1 whether the area be 20 sq. m. and the lift
coefficient 0'05, or the area 25 sq. m. and the lift coefficient 0'04, or
whatever be the combination. This being so, if the speed decreases to
18 m. per second, the square of which is 324, it is clear that the lift is
reduced from 400 to 324 kg.,- and consequently there is a reduction in
the lift of 76 kg. as stated.

If the normal speed were 30 m. per second, the product of the area and

the lift coefficient would be OArk = 0'444; the decrease in the speed to
900

28 m. per second (the square of which is 784) would give the lift a value
of 0-444 x 784 = 348 kg. The loss of lif t, therefore, would be only 52 kg.

THE EFFECT OF WIND ON AEROPLANES 193

duction of the lift coefficient by 0-005 would be 80 kg.
instead of 40 kg.

These results hold good irrespectively of the area and
the speed.

Finally, if both the speed and the angle of wind vary
at one and the same time, both results are added to one
another.

From this it may be deduced that for an aeroplane to
experience the least possible loss of lift owing to an atmo-
spheric disturbance, it should be light, fly at a high speed,
and possess a big lift coefficient.

These two latter conditions are not so contradictory as
might be supposed ; and if considered together, further
confirm the view expressed in Chapter III., as the result
of totally different considerations, that an increase in
the speed of aeroplanes should be sought for rather in
the reduction of their area than of their lift coefficient.
Apart from the question of weight, which will be dealt
with further on, this may be one of the reasons why, as
a general rule, monoplanes behave better in a wind than
biplanes.*

The relative action of a gust moving in the plane of
symmetry of an aeroplane, results, as we have just seen,
in a modification of the lift of the planes. This modifica-
tion produces the displacement effect.

Suppose, for instance, that an aeroplane flying hori-
zontally at a definite speed suddenly were to lose the
whole of its lift ; it would become comparable to a
projectile launched horizontally, and, while retaining a
certain forward speed, would fall. If the air in no way
resisted its fall, this would take place at the rate of any
body falling freely in a vacuum ; that is, after one second
it would have fallen about 5 m., at the end of 2 seconds
20 m., etc.

Its trajectory would be a curve bending ever more steeply

* Responsibility for this statement, in which I do not concur, rests
entirely in the author. — TRANSLATOR.

194 FLIGHT WITHOUT FORMULA

towards the earth. Naturally this curve would be flatter
the higher the flying speed of the aeroplane.

Actually the air opposes, in the vertical sense, con-
siderable resistance to the fall of a machine provided with
planes, so that an aeroplane would not fall so fast as men-
tioned above.

Moreover, as a gust is not instantaneous and only lasts
a short while, the flight-path straightens out again fairl}-
quickly as soon as the lift returns, and this the more quickly
the smaller the mass of the aeroplane.

This modification of the flight-path constitutes the dis-
placement effect due to the gust.

The pilot only feels, in the case under consideration, the
sensation of a vertical fall though actually this move-
ment is progressive. According to pilots' accounts these
vertical falls are considerable, from which one judges that
either the duration of the gusts is fairly long or that the
planes may, under given conditions, lose more than their
total lift.*

This displacement effect is devoid of danger, when it is
not excessive, if it is in the sense of raising the machine.
When it is considerable, the pilot corrects it by reducing
his incidence by means of the elevator.

On the other hand, if it tends to make the aeroplane
fall, it may be dangerous if occurring near the ground ; it
is here, moreover, that there always exists a source of
danger, for eddies are more frequent than higher up in
the atmosphere.

Besides, pilots always fear a loss of lift or, what is often
the equivalent, a loss of air speed, for, apart altogether

* The discovery made during the inquiry into certain accidents that
the upper stay-wires of monoplanes have broken in the air, would at first
sight appear to confirm the view that their wings may at times be struck
by the wind on their upper surface.

Nevertheless this view should be treated with caution, for the break-
age of the overhead stay-wires could be attributed equally well to the
effects of inertia produced when, at the end of a dive, the pilot flattens
out too abruptly.

THE EFFECT OF WIND ON AEROPLANES 195

from the ensuing fall, the aeroplane then flies in a con-
dition where the ordinary laws normally determining the
equilibrium and stability of an aeroplane no longer apply.
This stability may become most precarious, and this is
apparent to the pilot by the fact that the controls no longer
respond. The only remedy is to regain air speed, which
is effected by diving.*

Usually, therefore, the correction of displacement effects
due to gusts consists in diving. Nevertheless, if a head
gust slanting downward forced the aeroplane down, the
pilot would naturally have to elevate. In this case there
would be no loss of air speed, and the loss of lift would be
due to the reduction of the relative incidence.

Let us now turn to the rotary effects of atmospheric
disturbances acting in the plane of symmetry of the aero-
plane. A machine with a fixed elevator can only fly at a
single angle of incidence. Therefore, if the relative wind
which normally strikes an aeroplane changes its inclination
by reason of a gust, the machine will of its own accord
seek to resume, relatively to the new direction of the re-
lative wind, the only angle of incidence at which it flies
in longitudinal equilibrium.

The same thing will happen if the displacement effect
already referred to should modify the trajectory of the
centre of gravity ; the latter will always tend to adhere to
its flight-path.

The rotary effect resulting will take place all the quicker,
and will die away all the more rapidly, as the longitudinal
moment of inertia of the machine is smaller. Thus, in
the case, already considered, of an aeroplane losing air speed
and falling, it may do this bodily, without any appreciable
dive, if its moment of inertia is big ; whereas, if bow and
tail are lightly loaded, it yields to the gust and dives in a
more or less pronounced fashion.

* Air-speed indicators, consisting of some form of delicate anemo-
meter, constantly record the relative speed and enable the pilot to
operate his controls in good time.

196 PLIGHT WITHOUT FORMULA

This latter quality would appear to be the better one
of the two, since, in the case under consideration, the pilot
always has to dive to re-establish equilibrium. Hence, in
this respect, an aeroplane should have as small a longi-
tudinal moment of inertia as possible.

Another rotary effect may arise through a cause already
referred to — if the disturbance does not reach the main plane
and the tail simultaneously. In this case there is exerted
on the first surface struck, if considered independently from
the rest, a modification in the magnitude and the position
of the pressure, which in turn brings about a modification
in the couple which it normally exerted about the centre
of gravity.

If the couple due to the main plane takes the upper hand,
the machine tends to stall ; if the reverse takes place, it
tends to dive. A stalling aeroplane always loses some of
its air speed ; moreover, if the gust strikes it head-on the
machine is still further exposed, being stalled, to its dis-
turbing effect. As has already been shown, the correcting
movement for the majority of cases of displacement effect
consists not hi stalling but in diving.

For these various reasons, and excepting always the case
of a downward current forcing the machine down, the
rotary effect of a gust should cause the aeroplane to dive
of its own accord.

In this respect, the manner in which fore-and-aft balance
is maintained is most important. If the tail is a lifting tail
(see fig. 36, Chapter VI.), the pressure normally exerted on
the main plane passes in front of the centre of gravity.
This being so, the action of a gust striking the main plane
first, would produce as its rotary effect a stalling move-
ment, except only if the gust had a pronounced downward
tendency, in which case the stalling movement is the
right one.

A gust from the rear, striking the tail first, decreases its
lift and also provokes stalling. In every case, therefore,
the rotary effect of the gust is detrimental to stability.

THE EFFECT OF WIND ON AEROPLANES 197

A lifting tail which, as seen in Chapter VI., is the most
defective in regard to lateral stability in still air, is con-
sequently equally unfavourable in disturbed air.

On the other hand, if the tail is normally placed at a
negative angle (see fig. 42, Chapter VI.), the normal pressure
on the main plane passes behind the centre of gravity.
The action of a head gust, unless pointing downward to a
considerable extent, in this case produces as its rotary effect
a diving movement, and the same is true of a gust from
behind which diminishes the downward pressure normally
exerted on the tail plane. If the gust is a downward one
to a marked extent, it will tend to stall the machine, which,
again, is as it should be. In every case the rotary effect of
the gust is favourable.

The use of a negative tail plane, which has already been
seen to be excellent in regard to stability hi still air, is
therefore equally beneficial in disturbed air. Nor should
this cause surprise.

Previously it was shown that the presence of a plane
normally acting in front of the centre of gravity was
productive of longitudinal instability, since it really acted
as a reversed and overhung weathercock. It is quite clear
that if a gust strikes such a plane first, it will tend, being
a bad weathercock, to be displaced still further and thereby
become still more exposed to the disturbing action of
the gust.

On the other hand, if both the main plane and the tail
act behind the centre of gravity, where they combine to
procure for the machine an excellent degree of longitudinal
stability in still air, they will constitute a good weather-
cock which will always float in a head gust so that the
upsetting action vanishes,* and the aeroplane itself absorbs
the gust. In so far as gusts from behind are concerned,

* Earlier, it was stated (see p. 170) that longitudinally an aeroplane
must necessarily always be a bad weathercock, but some distinction of
quality still remains and, so far as the effects of wind are concerned, an
aeroplane should belong to a " good variety of bad weathercocks."

198 FLIGHT WITHOUT FORMULA

this arrangement is again productive of good stability
since the rotary effect due to the gust brings about the
very manoeuvre which the pilot would have otherwise to
perform in order to correct the displacement effect.

These rotary effects have an intensity and duration
depending on the moment of longitudinal inertia of the
machine. The science of mechanics proves that a definite
amount of disturbing energy applied to aeroplanes possess-
ing the same degree of longitudinal stability * gives them
an identical angular displacement irrespective of their
moment of inertia. The latter only affects the duration
of the displacement. The greater the moment of inertia,
the slower does the oscillation come about.

Nevertheless, it should be remembered that a force,
however great, can only put forth an amount of energy
proportional to the displacement produced.!

Hence, if the gust is only a brief one, the disturbing
energy applied to the aeroplane and the ensuing angular
displacement will be all the smaller the more reluctantly
the aeroplane yields to the gust. Wherefore, there is a
distinct advantage to be derived from increasing the
longitudinal moment of inertia.

But, if the gust lasts some considerable time, this ad-
vantage disappears and the great moment of inertia has
the effect of prolonging the disturbing impulse. Besides,
it may happen that two gusts follow one another at a
brief interval and that the second, which would encounter
an aeroplane with little inertia already re-established in
a position of equilibrium, would strike a machine heavily
loaded fore and aft before it had recovered, or even when
it was still under the influence of the first gust.

* In Chapter VI. it was shown that the longitudinal stability of an
aeroplane can be represented by the length of a pendulum arm weighted
at the end with the weight of the aeroplane.

f If a pony is harnessed to a heavy wagon, it will be unable to move
it ; its force will be wasted, since it will produce no energy. But if it
is harnessed to, a light cart, its force, though smaller than that put forth
in the former case, will produce useful energy.

THE EFFECT OF WIND ON AEROPLANES 199

Moreover, for the same reason, the first machine would
more readily answer its controls and would respond more
perfectly to the wishes of most pilots, who desire, above
all, a controllable aeroplane.

It should be noted that, in so far as rotary effects are
concerned, it is desirable that gusts should clear an aeroplane
as quickly as possible, and, for this reason, it should be
fairly short fore and aft, after the example of birds who
fly particularly well.

The negative-angle tail complies well with this require-
ment and also compensates the lessening of the lever arm of
the tail plane which ensues through its important increase
in stability due to the increase in the longitudinal V-

Moreover, by bringing the main and tail planes closer
together, the longitudinal moment of inertia is reduced,
whereby the machine is rendered more responsive to its
controls.

For these reasons, the author is of opinion that the
present type of aeroplane with its tail far outstretched
will give way to a machine at once much shorter, more
compact, and easier to control.*

Summarising our conclusions, we find that :

(1) In regard to the relative action of gusts, which are
the main cause of loss of equilibrium, an aeroplane should
be as light as possible, so as to be able to yield in the greatest
possible measure to the displacement effect of gusts, which
reduces their relative effect. This conclusion is clearly
open to question, and may be opposed by the illustration
that large ships have less to fear from a storm than small
boats. But the comparison is not exact, for the simple
reason that boats are supported by static means, whereas
aeroplanes are upheld in the air dynamically.

* Not that it will be possible to suppress the tail entirely, as some
have attempted to do. Oscillatory stability (see Chapter VII.) would
suffer if this were done, and the braking effect would disappear. Besides,
Nature would have made tailless birds, could these have dispensed with
their tails.

200 FLIGHT WITHOUT FORMULA

(2) Regarding its behaviour in a wind, an aeroplane
should :

(a) possess high speed, with the proviso that its speed
should not be obtained by reducing its lift co-
efficient, so that any increase in speed should
be achieved rather by reducing the area than
the lift coefficient ;
(6) be naturally stable longitudinally ;

(d) be short in the fore-and-aft dimension ;

(e) be so designed that any initial displacement due

to a gust causes it to turn head to the gust
instead of exposing it still further to its dis-
turbing effect.

The negative tail arrangement seems to answer the
most perfectly to (6), (c), (d), and (e).

It has often been stated that those provisions ensuring
stability in still air were harmful to stability in disturbed
air. If this were true, the future of aviation would indeed
be black. Fortunately it is erroneous, even though practice
has borne it out hitherto with few exceptions.

It has been attempted, as in the case of the brothers
Wright, to overcome this difficulty by only providing the
minimum degree of stability essential to the correct be-
haviour of a machine in still air, leaving the pilot to make
the necessary corrections to counteract the disturbing
effects of the wind by giving him exceptionally powerful
means of control.

The slight degree of natural stability possessed by such
an aeroplane renders it most responsive to its controls — a
feature agreeable to the majority of pilots. On the other
hand, by actuating the control a pilot may unduly modify,
even to a dangerous extent, the normal state of equilibrium.
More especially is this true of longitudinal equilibrium,
for here, as has been shown, a slight degree of stability
may change into actual instability — for instance, by putting
the elevator down too far. This is due (as explained in

THE EFFECT OF WIND ON AEROPLANES 201

Chapters VI. and VII.) to the fact that the sheaf of total
pressures of the aeroplane is thereby altered, with the result
that the longitudinal V is diminished, and consequently
the diminution of the angle of incidence, instead of increas-
ing stability, as in the case of advancing the centre of
gravity, would bring it down to vanishing-point.

Aeroplanes which display a tendency towards uncontrol-
lable dives, are simply momentarily unstable longitudinally
and refuse to answer the pilot's controls because, owing
to their acceleration, their dive soon becomes a headlong
fall, so that the precarious degree of stability which they
possessed in normal flight has disappeared. In such a case
it would be incorrect to say that an increase of speed
augments stability, for, on the contrary, when the speed
passes a certain limit termed the " critical speed " (in the
author's opinion, this term is not correct, since a well-
designed aeroplane should have no critical speed), all stability
vanishes.

An aeroplane should always be so designed as to be
naturally stable in still air, and at the same time every
effort should be made to arrange its structure so as to
render it stable also in disturbed air.

It has already been shown that it seems possible, in
regard to longitudinal stability, to achieve this result with-
out sacrificing controllability, which would appear to be
dependent, above all, on a small moment of inertia.

Whether the negative tail arrangement, previously re-
ferred to, or some other similar device should prove the
better in the long run, this for the time being is the right
road along which to make endeavours and to try to reduce
to the lowest possible degree the intervention of the pilot
in controlling the stability of an aeroplane. The whole
future of aviation is bound up in the solution of this problem.
An aeroplane should be able to fly in the worst weather
without demanding from its pilot an incessant, tiring, and
often dangerous struggle against the elements. Not until
this is achieved will aviation cease to be the sport of the

202 FLIGHT WITHOUT FORMULA

few and become a speedy and practical, and above all, safe,
means of locomotion.

It has ere now been sought to reduce the necessity for
constant control on the part of the pilot by rendering
aeroplanes automatically stable. The problem is an un-
usually complex one, for automatic stability devices are
required to correct not only the effects of gusts that come
from without, but faults that arise from within the aero-
plane itself, such as a loss of power, motor failure, mis-
takes in piloting, etc.

This being so, if a device of this nature fulfils one part'
of its required functions, almost inevitably it will fail in
others, and this is the rock against which all attempts so
far made have been shattered. Not that the difficulty
cannot be overcome, but it is undoubtedly a grave one.

Hitherto such attempts at solution as have been made
have usually related to longitudinal stability. Among such
devices may be mentioned the ingenious invention of
M. Doutre, who utilised the effects of inertia exerted on
weights to actuate, at any change of air speed, the elevator
through the intermediary of a servo-motor.

Even now some lessons may be drawn from previous
attempts. More especially would it seem desirable to
prevent the effects of gusts rather than to correct them
once they have been produced. The use of " antenna " or
" feelers " — that is, of some kind of organ instantaneously
yielding to aerial disturbances and thus preparing, through
the intermediary of the requisite controls, an aeroplane to
meet the gust — would seem preferable to organs which only
right it once it has assumed an inclined position after
having been struck by the gust.

Important results, in this respect, also appear to have
been obtained by M. Moreau, who seems to have succeeded
in applying the principle of the pendulum to produce a
self-righting device.

In addition it has been sought to ensure automatic
stability by constantly maintaining the air speed of an

THE EFFECT OF WIND ON AEROPLANES 203

aeroplane. But it has already been shown that this is
inadequate in certain circumstances, more especially if the
aeroplane has a small lift coefficient, which is the case with
machines of large wing area, and the lift often decreases
to a far greater extent as the result of a decrease hi the
relative incidence than in the speed. Hence, not only the
relative speed, but the relative incidence should be preserved.

In regard to the effects of wind alone, therefore, the
problem is already complicated enough ; but it becomes
even more complex if disturbances due to the machine
itself are taken into consideration.

Without the slightest wish to deny the great importance
of the problem, the author, nevertheless, reiterates his
opinion that the first necessity is to so design the structure
of an aeroplane as to render it immune from dangers through
wind. Later, an automatic stability device could be
added in order to correct in just proportion the effects of
gusts and further to correct disturbances due to the machine
itself. If this were done, the functions of automatic stability
devices would be greatly simplified.

In addition, it should not be forgotten that by adding to
an aeroplane further moving organs which are consequently
subject to lagging and even to breakdowns, an element of
danger is created. In any event, any such device must
perforce constitute a complication.*

Until now only those gusts have been considered which
blow in the plane of symmetry of the aeroplane — straight
gusts which only affect the flying and longitudinal equili-

* Virtually, this stricture, while perfectly correct in itself, only applies
to such extraneous stability devices as those of Doutre and Moreau, and
not to the automatic stability inherent in the forms of the aeroplane
itself which has been produced by J. W. Dunne. This latter possesses
automatic stability in both senses, and in principle is based on the auto-
matic maintenance of air speed without the pilot's intervention. In
this respect it undoubtedly constitutes one of the greatest advances yet
made in aviation, though opinions may well differ on the point whether
it is desirable to rob the pilot of control in order to confide it to automatic
mechanism. — TRANSLATOR.

204

FLIGHT WITHOUT FORMULA

brium of the machine. Let us now examine the effect
of side-gusts. By doing so, we shall have considered
the effect of almost every variety of aerial disturbance,
which can in most cases be resolved into an action directed
in the plane of symmetry of the aeroplane and into one
acting laterally.

In this case again we distinguish a primary effect and a
relative effect.

If the aeroplane had no inertia, it would immediately be

FIG. 84.

carried away by the gust together with the mass of sup-
porting air, and this movement would not be perceptible
to the pilot except by observing the ground beneath. But
this is a purely hypothetical case, and the gust exerts
a relative action on the machine, which is the more pro-
nounced the greater the mass of the latter.

This action is perceptible by a modification in the
direction and speed of the relative wind. For, if the
aeroplane were flying in still air, thereby encountering a
relative wind GA (fig. 84), and were struck by a lateral
gust whose action is represented by the speed GB, the
relative speed of the machine becomes GC. Both the

THE EFFECT OF WIND ON AEROPLANES 205

magnitude and direction of the speed have, consequently,
altered.

The fact of the relative speed varying in magnitude
shows that, apart from effects due to its dissymmetrical
position, the relative action of a side-gust must exert on
the flying and longitudinal equilibrium an influence similar
to that produced by the straight gusts already considered.

A lateral gust, therefore, can cause an aeroplane to rise or
fall at the same tune that it disturbs its longitudinal equili-
brium. But for the sake of simplicity this part of the
effects of side-gusts may be ignored, and only those effects
need be taken into account which modify the direction
of flight, and lateral and directional stability.

First, the displacement effect due to the relative action of
a side-gust consists in creating a centripetal force tending
to curve the flight-path and to produce a turn in the direction
opposite to that from which the gust comes.

Among the rotary effects, as in the case of straight gusts,
that particular one should first be distinguished which
causes an aeroplane, in regard to directional equilibrium,
to adhere to its flight-path — or, in other words, to behave
like a good weathercock.

If the flight -path curves, as the result of the displacement
effect of a gust, in the opposite direction to that from which
the gust comes, the rotary effect which will tend to make
the aeroplane adhere to its new flight-path will cause it to
be exposed still further to the disturbing effect of the gust.
It will turn away from the wind. So far as this point is
concerned, it would seem desirable that an aeroplane should
take up its new flight-path — as slowly as possible.

But a second rotary effect causes the aeroplane to assume
the new direction of the relative wind, like a good weather-
cock, and this is an advantage, since, by heading into the
wind, the lateral disturbing effect of the gust is damped out.

Of these two rotary effects the second is probably the
first to occur and to remain the more intense.

In order to reduce the first rotary effect, the lateral

206 FLIGHT WITHOUT FORMULA

resistance of the aeroplane — that is, its keel surface — should
not exceed certain proportions. Moreover, the directional
stability should also be reduced to a minimum from this
point of view ; the second and more important rotary effect,
on the other hand, points to an increase in directional
stability as desirable.

Both theories have their friends and foes, and here again
the view has been advanced that the aeroplane should be
given only that measure of stability which is strictly
necessary in order to prevent it from yielding too easily
to the rotary effects of gusts and to render it easily
controllable. Such a reduction in directional stability
is not so detrimental as a diminution of longitudinal
stability, since it in no way affects the cardinal principles
of sustentation.

Nevertheless, in the author's opinion a definite degree of
directional stability is desirable, since this would also
produce some amount of lateral stability which is always
somewhat defective. In any case, usually the structure of
the aeroplane and the rudder in the rear suffice for the
purpose.

There remain the most important rotary effects due to
side-gusts — those which affect lateral stability.

Any modification in the direction of the relative wind
results in a lateral displacement of the normal pressure on
the main planes, which causes a couple tending to tilt the
aeroplane sideways. If that wing which is struck by the
gust rises, the aeroplane will turn into the opposite direction,
thus turning away from the wind, and thereby, as already
seen, exposes itself still further to the disturbing effect of
the gust.

But if the wing struck by the gust falls, the aeroplane
swings round, heading into the wind, which damps out the
disturbing effect. These movements are intensified by
reason of the gust not striking both wings at once.

According to the principle already cited, the initial
displacement due to a gust should cause an aeroplane to

THE EFFECT OF WIND ON AEROPLANES 207

turn into the wind instead of causing it to become exposed
to the disturbing influence still further, which renders the
second rotary effect the more favourable.

If the wings are straight and, still more, if they have a
lateral dihedral or V, the first effect is produced. Hence
a lateral dihedral seems unfavourable in disturbed air.
Besides, it is fast disappearing, and pilots of such machines
are obliged to counteract the effects of gusts by lowering
the wing struck first — that is, of momentarily suppressing,
as far as is in their power, the lateral dihedral, while swinging
round into the wind.

On the other hand, if the wings have an invertefl dihedral
or A> the rotary effect of a side-gust will be the second
and desirable effect ; the aeroplane will turn into the
wind of its own accord, which will cause the disturbing
effect to disappear.

Captain Ferber from the very first pointed out this fact
and remarked that sea-birds only succeeded in gliding in
a gale because they placed their wings so as to form an
inverted dihedral angle. But he also thought that these
birds could only assume this attitude, believed by him to
be unstable, by constant balancing. In Chapters VIII. and
IX. it was shown that it is possible, by lower ing the rolling
axis of an aeroplane in front (by lowering the centre of
gravity, or better, by raising the tail), to build machines
with wings forming a downward dihedral and nevertheless
stable in still air.*

In regard to lateral stability, as with longitudinal, the
natural stability of an aeroplane and good behaviour in a
wind are, contrary to general opinion, in no wise incom-
patible, and both these important qualities can be obtained
in one and the same machine by a suitable arrangement of
its parts.

* As previously mentioned, the " Tubavion " monoplane has flown
with its wings so arranged, and the pilot is stated to have noted a great
improvement in its behaviour in winds. This machine had a low centre
of gravity and a high tail.

208 FLIGHT WITHOUT FORMULA

Attempts to produce automatic lateral stabilisers have
hitherto not given very good results.*

So far as the moment of rolling inertia is concerned,
previous considerations point to the desirability of reducing
this as much as possible by the concentration of masses.
The machine is thus rendered easily controllable, and the
rapidity of its oscillations guards against the danger aris-
ing from too quick a succession of two gusts. This is
of exceptional importance from the point of view of lateral
stability, which we know to be the least effective of all or,
at any rate, the most difficult to obtain in any marked

Summarising these conclusions, it may be stated, that
for good behaviour in winds, an aeroplane should :

(1) be light, thus yielding more readily to the primary

effect of gusts, whereby it is not so much affected by
their relative action ; only if this relative action
could be wholly eliminated would an increase in the
weight become an advantage ;

(2) fly normally at high speed, provided that an increase

in speed be not obtained by unduly reducing the lift
coefficient ;

(3) be naturally stable both longitudinally and laterally ;

(4) have a small moment of inertia and its masses con-

centrated ;

(5) head into the wind instead of turning away from it.
The fulfilment of the last condition is the most likely to

produce the best results in regard to the behaviour of an
aeroplane in a wind, and this has been shown to be in
no way incompatible with excellent stability in still air
and adequate controllability. The arrangement proposed
by the author — a negative-angle tail and a downward

* This is hardly correct so far as the Dunne aeroplane is concerned,
which is automatically stable in a wind. This machine, it should be
noted, has in effect a downward dihedral and a comparatively low centre
of gravity, coupled with a relatively high tail which is constituted by the
wing-tips. — TRANSLATOR.

THE EFFECT OF WIND ON AEROPLANES 209

dihedral* — is not perhaps that which careful experiment
methodically pursued would finally cause to be adopted ;
but at any rate it provides a good starting-point.

What is required first of all is to so design the structure
itself of the aeroplane as to render it immune to danger
from gusts. The future of aviation depends upon this to
a large extent, and it is for this reason that attention has
been drawn to it with such insistence in these pages, for
in this respect much, if not almost all, remains to be done.

Afterwards, may come the study of movable organs
producing automatic stability, and in all probability this
study will have been greatly simplified if the first essential
condition has been complied with.

Who knows whether one day we shall not learn how to
impress into our service, like the birds, that very internal
work of the wind which now constitutes a source of danger
and difficulty ? Some species of birds appear to know
the secret of how to utilise the external energy of the
movements of the atmosphere and to remain aloft hi the
air for hours at a time without expending the slightest
muscular effort.

It is certain that for this purpose they make use of ascend-
ing currents, but it is difficult to believe that these currents
are sufficiently permanent to explain the mode of soaring
flight alluded to.

More probable is it that birds which practise soaring flight
— be it noted that they are all large birds, and consequently
possessing considerable inertia — meeting a head gust, give
their wings a large angle of incidence and thus rise upon
the gust, and then glide down at a very flat angle in the
ensuing lull.

Even in our latitudes certain big birds of prey, such as
the buzzard, rise up into the air continuously, without any
motion of then- wings, but always circling, when the wind

* This arrangement was first proposed by the author in a paper con-
tributed to the Academic des Sciences on March 25, 1911 (Comples Bendus,
voL clii. p. 1295).

210 FLIGHT WITHOUT FORMULAE

is strong enough. This circling appears essential, and may
possibly be explained on the supposition that the circling
speed is in some way connected with the rhythmic wave-
like pulsations of the atmosphere in such a fashion that
these pulsations, whether increasing or diminishing, are
always met by the bird as increasing pulsations, and on
this account it circles.

It appears in no way impossible that we should one day
be able to imitate the birds and to remain, without expend-
ing power, in the air on such days when the intensity of
atmospheric movements, an inexhaustible supply of power,
is sufficient for the purpose.

One thing is to be remembered : wind, and probably
irregular wind, is absolutely essential to enable such flight
to be possible ; it would be an idle dream to hope to over-
come the never-failing force of gravity without calling
into play some external forces of energy, and on those
days when this energy could not be derived from the wind,
it would have to be supplied by the motor.

But in any event this stage has not yet been reached,
and before we attempt to harness the movements of the
atmosphere they must no longer give cause for fear. To
this end the aerial engineer must direct all his efforts for
the present.

The really high-speed aeroplane forms one solution, even
though probably not the best, since such machines must
always remain dangerous in proximity to the surface of
the earth.*

Without a doubt, a more perfect solution awaits us some-
where, and the future will surely bring it forth into the
light. On that day the aeroplane will become a practical
means of locomotion.

Let the wish that this day may come soon conclude this

* Slowing up preparatory to alighting forms no solution to the difficulty,
since the machine would lose those very advantages, conferred by its
high speed, precisely at the moment when these were most needed, in the
disturbed lower air.

THE EFFECT OF WIND ON AEROPLANES 211

work. Every effort has been made to render the chapters
that have gone before as simple and as attractive as the
subject, often it is to be feared somewhat dry, permitted.

Not a single formula has been resorted to, and if the
author has succeeded in his task of rendering the under-
standing of his work possible with the simple aid of such
knowledge as is acquired at school, this is mainly due to
the distinguished research work which has lately furnished
aeronautical science with a mass of valuable facts : to the
work of M. Eiffel, to which reference has so often been
made in the foregoing pages.

No more fitting conclusion to these chapters could there-
fore be devised than this slight tribute to the indefatigable
zeal and the distinguished labours of this great scientific
worker who has rendered this book possible.

Printed by T. and A. CONSTABLE, Printers to His Majesty
at the Edinburgh University Press, Scotland

UNIVERSITY OF CALIFORNIA LIBRARY

Los Angeles
This book is DUE on the last date stamped below.

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COt QBi

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NOV 17*969

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